On Some Quadratic Algebras I $\frac{1}{2}$: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials

We study some combinatorial and algebraic properties of certain quadratic algebras related with dynamical classical and classical Yang-Baxter equations. One can find more details about the content of present paper in Extended Abstract.


Extended Abstract
We introduce and study a certain class of quadratic algebras, which are nonhomogenious in general, together with the distinguish set of mutually commuting elements inside of each, the so-called Dunkl elements. We describe relations among the Dunkl elements in the case of a family of quadratic algebras corresponding to a certain splitting of the universal classical Yang-Baxter relations into two three term relations. This result is a further extension and generalization of analogous results obtained in [26], [76] and [51]. As an application we describe explicitly the set of relations among the Gaudin elements in the group ring of the symmetric group, cf [71]. We also study relations among the Dunkl elements in the case of (nonhomogeneous) quadratic algebras related with the universal dynamical classical Yang-Baxter relations. Some relations of results obtained in papers [26], [52], [47] with those obtained in [35] are pointed out. We also identify a subalgebra generated by the generators corresponding to the simple roots in the extended Fomin-Kirillov algebra with the DAHA, see Section 4.3.
The set of generators of algebras in question, naturally corresponds to the set of edges of the complete graph K n (to the set of edges and loops of the complete graph with loops K n in dynamical case). More generally, starting from any subgraph Γ of the complete graph with loops K n we define a (graded) subalgebra 3T (0) n (Γ) of the (graded) algebra 3T (0) n ( K n ) [44]. In the case of loop-less graphs Γ ⊂ K n we state Conjecture which relates the Hilbert polynomial of the abelian quotient 3T  n (Γ) and the chromatic polynomial of the graph Γ we started with. We check our Conjecture for the complete graphs K n and the complete bipartite graphs K n,m . Besides, in the case of complete multipartite graph K n 1 ,...,nr , we identify the commutative subalgebra in the algebra 3T (0) N (K n 1 ,...,nr ), N = n 1 + · · · + n r , generated by elements θ (N ) j,k j := e k j (θ (N ) N j−1 +1 , . . . , θ (N ) N j ), 1 ≤ j ≤ r, 1 ≤ k j ≤ n j , N j := n 1 + . . . + n j , N 0 = 0, with the cohomology ring H * (Fl n 1 ,...,nr , Z) of the partial flag variety Fl n 1 ,...,nr . In other words, the set of (additive) Dunkl  N j } plays a role of the K-theoretic version of Chern roots of the tautological vector bundle ξ j over the partial flag variety Fl n 1 ,...,nr . As a byproduct for a given set of weights ℓ = {ℓ ij } 1≤i<j≤r we compute the Tutte polynomial T (K We also introduce and study a family of (super) 6-term relations algebras, and suggest a definition of " multiparameter quantum deformation " of the algebra of the curvature of 2-forms of the Hermitian linear bundles over the complete flag variety Fl n . This algebra can be treated as a natural generalization of the (multiparameter) quantum cohomology ring QH * (Fl n ), see Section 4.2.
Yet another objective of our paper is to describe several combinatorial properties of some special elements in the associative quasi-classical Yang-Baxter algebra [47], including among others the so-called Coxeter element and the longest element. In the case of Coxeter element we relate the corresponding reduced polynomials introduced in [90], with the β-Grothendieck polynomials [27] for some special permutations π (n) k . More generally, we identify the β-Grothendieck polynomial G (β) π (n) k (X n ) with a certain weighted sum running over the set of k-dissections of a convex (n + k + 1)-gon. In particular we show that the specialization G π (n) k (X n ) counts the number of k-dissections of a convex (n + k + 1)-gon according to the number of diagonals involved. When the number of diagonals in a k-dissection is the maximal possible (equals to n(2k − 1) − 1), we recover the well-known fact that the number of k-triangulations of a convex (n + k + 1)-gon is equal to the value of a certain Catalan-Hankel determinant, see e.g. [85].
We also show that for a certain 5-parameters family of vexillary permutations, the specialization x i = 1, ∀i ≥ 1, of the corresponding β-Schubert polynomials S (β) w (X n ) turns out to be coincide either with the Fuss-Narayana polynomials and their generalizations, or with a (q, β)deformation of V SASM or that of CST CP P numbers, see Corollary 5.2, (B).. As examples we show that (a) the reduced polynomial corresponding to a monomial x n 12 x m 23 counts the number of (n, m)-Delannoy paths according to the number of N E-steps, see Lemma 5.2;(b) if β = 0, the reduced polynomial corresponding to monomial (x 12 x 23 ) n x k 34 , n ≥ k, counts the number of of n up, n down permutations in the symmetric group S 2n+k+1 , see Proposition 5.9; see also Conjecture 18. We also point out on a conjectural connection between the sets of maximal compatible sequences for the permutation σ n,2n,2,0 and that σ n,2n+1,2,0 from one side, and the set of V SASM (n) and that of CST CP P (n) correspondingly, from the other, see Comments 5.7 for details. Finally, in Section 5.1.1 we introduce and study a multiparameter generalization of reduced polynomials introduced in [90], as well as that of the Catalan, Narayana and (small) Schröder numbers.
In the case of the longest element we relate the corresponding reduced polynomial with the Ehrhart polynomial of the Chan-Robbins-Yuen polytope, see Section 5.3. More generally, we relate the (t, β)-reduced polynomial corresponding to monomial

Introduction
The Dunkl operators have been introduced in the later part of 80's of the last century by Charles Dunkl [21], [22] as a powerful mean to study of harmonic and orthogonal polynomials related with finite Coxeter groups. In the present paper we don't need the definition of Dunkl operators for arbitrary (finite) Coxeter groups, see e.g. [21], but only for the special case of the symmetric group S n . Definition 1.1. Let P n = C[x 1 , . . . , x n ] be the ring of polynomials in variables x 1 , . . . , x n . The type A n−1 (additive) rational Dunkl operators D 1 , . . . , D n are the differential-difference operators of the following form
The key property of the Dunkl operators is the following result.
Another fundamental property of the Dunkl operators which finds a wide variety of applications in the theory of integrable systems, see e.g. [36], is the following statement: the operator For example, the type A n−1 rational truncated Dunkl operator has the following form Clearly the truncated Dunkl operators generate a commutative algebra. The important property of the truncated Dunkl operators is the following result discovered and proved by C.Dunkl [22]; see also [4] for a more recent proof. Theorem 1.2. (C.Dunkl [22], Y.Bazlov [4]) For any finite Coxeter group (W, S) the algebra over Q generated by the truncated Dunkl operators D 1 , . . . , D l is canonically isomorphic to the coinvariant algebra A W of the Coxeter group (W, S).
Recall that the elementary symmetric polynomials e i (X n ), i = 1, . . . , n, are defined through the generating function where we set X n := (x 1 , . . . , x n ). It is well-known that in the case W = S n , the isomorphism (1.2) can be defined over the ring of integers Z. • Cohomology and K-theory rings of affine flag varieties ?
• Complex reflection groups ? The present paper is an extended Introduction to a few items from Section 5 of [47].
The main purpose of my paper "On some quadratic algebras, II" is to give some partial answers on the above questions basically in the case of the symmetric group S n .
The purpose of the present paper is to draw attention to an interesting class of nonhomogeneous quadratic algebras closely connected (still mysteriously !) with different branches of Mathematics such as Classical  What we try to explain in [47] is that upon passing to a suitable representation of the quadratic algebra in question, the subjects mentioned above, are a manifestation of certain general properties of that quadratic algebra.
From this point of view, we treat the commutative subalgebra generated by the additive (resp. multiplicative) truncated Dunkl elements in the algebra 3T n (β), see Definition 3.1, as universal cohomology (resp. universal K-theory) ring of the complete flag variety Fl n . The classical or quantum cohomology (resp. the classical or quantum K-theory) rings of the flag variety Fl n are certain quotients of that universal ring.
For example, in [50] we have computed relations among the (truncated) Dunkl elements {θ i , i = 1, . . . , n} in the elliptic representation of the algebra 3T n (β = 0). We expect that the commutative subalgebra obtained is isomorphic to elliptic cohomology ring ( not defined yet, but see [33] , [32]) of the flag variety Fl n .
Another example from [47]. Consider the algebra 3T n (β = 0). One can prove [47]  One can check that upon passing to the elliptic representation of the algebra 3T n (β = 0), see Section 3.1, or [47], Section 5.1.7, or [50], for the definition of elliptic representation, the above identities (A) and (B) finally end up correspondingly, to be the Summation formula and the N = 1 case of the Duality transformation formula for multiple elliptic hypergeometric series (of type A n−1 ), see e.g. [41] , or Appendix V, for the explicit forms of the latter. After passing to the so-called Fay representation [47], the identities (A) and (B) become correspondingly to be the Summation formula and Duality transformation formula for the Riemann theta functions of genus g > 0, [47]. These formulas in the case g ≥ 2 seems to be new. Worthy to mention that the relation (A) above can be treated as a "non-commutative analogue" of the well-known recurrence relation among the Catalan numbers. The study of "descendent relations" in the quadratic algebras in question was originally motivated by the author attempts to construct a monomial basis in the algebra 3T A few words about the content of the present paper. Example 1.1 can be viewed as an illustration of the main problems we are treaded in Sections 2 and 3 of the present paper, namely the following ones.
• Let {u ij , 1 ≤ i, j ≤ n} be a set of generators of a certain algebra over a commutative ring K. The first problem we are interested in is to describe "a natural set of relations" among the generators {u ij } 1≤i,j≤n which implies the pair-wise commutativity of dynamical Dunkl elements • Should this be the case then we are interested in to describe the algebra generated by "the integrals of motions", i.e. to describe the quotient of the algebra of polynomials K[y 1 , . . . , y n ] by the two-sided ideal J n generated by non-zero polynomials F (y 1 , . . . , y n ) such that F (θ 1 , . . . , θ n ) = 0 in the algebra over ring K generated by the elements {u ij } 1≤i,j≤n .
• We are looking for a set of additional relations which imply that the values of elementary symmetric polynomials e k (y 1 , . . . , y n ), 1 ≤ k ≤ n, on the Dunkl elements θ (n) 1 , . . . , θ (n) n do not depend on the variables {u ij , 1 ≤ i = j ≤ n}. If so, one can defined deformation of elementary symmetric polynomials, and make use of it and the Jacobi-Trudi formula, to define deformed Schur functions, for example. We try to realize this program in Sections 2 and 3.
In Section 2, see Definition 2.2, we introduce the so-called dynamical classical Yang-Baxter algebra as "a natural quadratic algebra" in which the Dunkl elements form a pair-wise commuting family. It is the study of the algebra generated by the (truncated) Dunkl elements that is the main objective of our investigation in [47] and the present paper. In subsection 2.1 we describe few representations of the dynamical classical Yang-Baxter algebra DCY B n related with • quantum cohomology QH * (Fl n ) of the complete flag variety Fl n , cf [25]; • quantum equivariant cohomology QH * T n ×C * (T * Fl n ) of the cotangent bundle T * Fl n to the complete flag variety, cf [35]; • Dunkl-Gaudin and Dunkl-Uglov representations, cf [71], [94]. In Section 3, see Definition 3.1, we introduce the algebra 3HT n (β), which seems to be the most general (noncommutative) deformation of the (even) Orlik-Solomon algebra of type A n−1 , such that it's still possible to describe relations among the Dunkl elements, see Theorem 3.1.
As an application we describe explicitly a set of relations among the (additive) Gaudin / Dunkl elements, cf [71].
◮◮ It should be stressed at this place that we treat the Gaudin elements/operators (either additive or multiplicative) as images of the universal Dunkl elements/operators (additive or multiplicative) in the Gaudin representation of the algebra 3HT n (0). There are several other important representations of that algebra, for example, the Calogero-Moser, Bruhat, Buchstaber-Felder-Veselov (elliptic), Fay trisecant (τ -functions), adjoint, and so on, considered (among others) in [47]. Specific properties of a representation chosen 3 (e.g. Gaudin representation) imply some additional relations among the images of the universal Dunkl elements (e.g. Gaudin elements) should to be unveiled. ◭◭ We start Section 3 with definition of algebra 3T n (β) and its "Hecke" 3HT n (β) and "elliptic" 3M T n (β) quotients. In particular we define an elliptic representation of the algebra 3T n (0), [50], and show how the well-known elliptic solutions of the quantum Yang-Baxter equation due to A. Belavin and V. Drinfeld, see e.g. [5], S. Shibukawa and K. Ueno [86], and G. Felder and V.Pasquier [24], can be plug in to our construction, see Section 3.1.
In Subsection 3.2 we introduce a multiplicative analogue of the the Dunkl elements {Θ j ∈ 3T n (β), 1 ≤ j ≤ n} and describe the commutative subalgebra in the algebra 3T n (β) generated by multiplicative Dunkl elements [51]. The latter commutative subalgebra turns out to be isomorphic to the quantum equivariant K-theory of the complete flag variety Fl n [51].
In Subsection 3.3 we describe relations among the truncated Dunkl-Gaudin elements. In this case the quantum parameters q ij = p 2 ij , where parameters {p ij = (z i − z j ) −1 , 1 ≤ i < j ≤ n} satisfy the both Arnold and Plücker relations. This observation has made it possible to describe a set of additional rational relations among the Dunkl-Gaudin elements, cf [71]. 3 For example, in the cases of either Calogero-Moser or Bruhat representations one has an additional constraint, namely, u 2 ij = 0 for all i = j. In the case of Gaudin representation one has an additional constraint u 2 ij = p 2 ij , where the (quantum) parameters {pij = 1 x i −x j , i = j}, satisfy simultaneously the Arnold and P lücker relations, see Section 2, (II). Therefore, the (small) quantum cohomology ring of the type An−1 full flag variety Fln and the Bethe subalgebra(s) (i.e. the subalgebra generated by Gaudin elements in the algebra 3HTn(0)) correspond to different specializations of " quantum parameters" {qij := u 2 ij } of the universal cohomology ring (i.e. the subalgebra/ring in 3HTn(0) generated by (universal) Dunkl elements). For more details and examples, see Section 2.1 and [47].
In Subsection 3.4 we introduce an equivariant version of multiplicative Dunkl elements, called shifted Dunkl elements in our paper, and describe (some) relations among the latter. This result is a generalization of that obtained in Section 3.1 and [51]. However we don't know any geometric interpretation of the commutative subalgebra generated by shifted Dunkl elements.
◮◮ An analog of the algebras 3T n and 3T (β) n , 3HT n , etc treated in the present paper, can be defined for any (oriented or not) matroid M. We denote these algebras as 3T (M) and 3T (β) (M). One can show (A.K.) that the abelianization of the algebra 3T (β) (M), denoted by 3T (β) (M) ab , is isomorphic to the Gelfand-Varchenko algebra corresponding to a matroid M, whereas the algebra 3T (β=0) (M) ab is isomorphic to the (even) Orlik-Solomon algebra OS + (M) of a matroid M 6 . We consider and treat the algebras 3T (M), 3HT (M),.... as equivariant noncommutative (or quantum) versions of the (even) Orlik-Solomon algebras associated with matroid (including hyperplane, graphic, ... arrangements). However a meaning of a quantum deformation of the (even or odd) Orlik-Solomon algebra suggested in the present paper, is missing, even for the braid arrangement of type A n . Generalizations of the Gelfand-Varchenko algebra has been suggested and studied in [45], [47] and in the present paper under the name quasi-associative Yang-Baxter algebra, see Section 5. ◭◭ In the present paper we basically study the abelian quotient of the algebra 3T (0) n (Γ), where graph Γ has no loops and multiple edges, since we expect some applications of our approach to the theory of chromatic polynomials of planar graphs, in particular to the complete multipartite graphs K n 1 ,...,nr and the grid graphs G m,n 7 . Our main results hold for the complete multipartite, cyclic and line graphs. In particular we compute their chromatic and Tutte polynomials, see Proposition 4.2 and Theorem 4.3. As a byproduct we compute the Tutte polynomial of the ℓweighted complete multipartite graph K (ℓ) n 1 ,...,nr where ℓ = {ℓ ij } 1≤i<j≤r , is a collection of weights, i.e. a set of non-negative integers.
More generally, for a set of variables {{q ij } 1≤i<j≤n , x, y} we define universal Tutte polynomial T n ({q ij }, x, y) ∈ Z[q ij ][x, y] such that for any collection on non-negative integers {m ij } 1≤i<j≤n and a subgraph Γ ⊂ K (m) n of the complete graph K n with each edge (i, j) comes with multiplicity m ij , the specialization of the universal Tutte polynomial T n ({q ij }, x, y) is equal to the Tutte polynomial of graph Γ multiplied by the factor (t − 1) κ(Γ) : Here and after κ(Γ) demotes the number of connected components of a graph Γ. In other words, one can treat the universal Tutte polynomial T n ({q ij }, x, y) as a "reproducing kernel" for 4 Independently the algebra 3T (0) n (Γ) has been studied in [9], where the reader can find some examples and conjectures. 5 To avoid confusions, it must be emphasized that the defining relations for algebras 3Tn(Γ) and 3Tn(Γ) (0) may have more then three terms. 6 For a definition and basic properties of the Orlik-Solomon algebra corresponding to a matroid see e.g, Y. Kawahara, On Matroids and Orlik-Solomon Algebras Annals of Combinatorics 8 (2004) 63-80.
7 See e.g. wolf ram.com/GridGraph.htm for a definition of grid graph Gm,n the Tutte polynomials of all graphs with the number of vertices not exceeded n.
We also state Conjecture 4.2 that for any loopless graph Γ (possibly with multiple edges) the algebra 3T (0) |Γ| (Γ) ab is isomorphic to the even Orlik-Solomom algebra OS + (A Γ ) of the graphic arrangement associated with graph Γ in question. At the end we emphasize that the case of the complete graph Γ = K n reproduces the results of the present paper and those of [47], i.e. the case of the full flag variety Fl n . The case of the complete multipartite graph Γ = K n 1 ,...,nr reproduces the analogue of results stated in the present paper for the case of full flag variety Fl n , to the case of the partial flag variety F n 1 ,...,nr , see [47] for details. In Section 4.1.3 we sketch how to generalize our constructions and some of our results to the case of the Lie algebras of classical types 8 .
In Section 4. 2 we briefly overview our results concerning yet another interesting family of quadratic algebras, namely the six-term relations algebras 6T n , 6T (0) n and related ones. These algebras also contain a distinguished set of mutually commuting elements called Dunkl elements {θ i , i = 1, . . . , n} given by θ i = j =i r ij , see Definition 4.10.
In Subsection 4.2.2 we introduce and study the algebra 6T ⋆ n in greater detail. In particular we introduce a "quantum deformation" of the algebra generated by the curvature of 2-forms of of the Hermitian linear bundles over the flag variety Fl n , cf [78].
In Subsection 4.2.3 we state our results concerning the classical Yang-Baxter algebra CY B n and the 6-term relation algebra 6T n . In particular we give formulas for the Hilbert series of these algebras. These formulas have been obtained independently in [3] The paper just mentioned, contains a description of a basis in the algebra 6T n , and much more.
In Subsection 4.2.4 we introduce a super analog of the algebra 6T n , denoted by 6T n,m , and compute its Hilbert series.
Finally, in Subsection 4.3 we introduce extended nil-three term relations algebra 3T n and describe a subalgebra inside of it which is isomorphic to the double affine Hecke algebra of type A n−1 , cf [15].
In Section 5 we describe several combinatorial properties of some special elements in the associative quasi-classical Yang-Baxter algebra 9 , denoted by ACY B n . The main results in that direction were motivated and obtained as a by-product, in the process of the study of the the structure of the algebra 3HT n (β). More specifically, the main results of Section 5 were obtained in the course of "hunting for descendant relations" in the algebra mentioned, which is an important problem to be solved to construct a basis in the nil-quotient algebra 3T (1) Let w ∈ S n be a permutation, consider the specialization In other words, the polynomial R w (q, β) has non-negative integer coefficients 10 . For late use we define polynomials (2) Let w ∈ S n be a permutation, consider the specialization Then (3) Let w be a permutation, then For the reader convenience we collect some basic definitions and results concerning the β-Grothendieck polynomials in Appendix I.
Let us observe that R w (1, 1) = S w (1), where S w (1) denotes the specialization x i := 1, ∀i ≥ 1, of the Schubert polynomial S w (X n ) corresponding to permutation w. Therefore, R w (1, 1) is equal to the number of compatible sequences [8] (or pipe dreams, see e.g. [85] ) corresponding to permutation w.
Let w ∈ S n be a permutation and l := ℓ(w) be its length. Denote by CS(w) = {a = (a 1 ≤ a 2 ≤ · · · ≤ a l ) ∈ N l } the set of compatible sequences [8] corresponding to permutation w.
• Define statistics r(a) on the set of all compatible sequences CS n := w∈Sn CS(w) in a such way that a∈CS(w) q a 1 β r(a) = R w (q, β).
• Find a geometric interpretation, and investigate combinatorial and algebra-geometric proper- where for a permutation w ∈ S n we denoted by S (β) w (X n ) the β-Schubert polynomial defined as follows We expect that polynomial S (β) w (1) coincides with the Hilbert polynomial of a certain graded commutative ring naturally associated to permutation w. Remark 1.1. It should be mentioned that, in general, the principal specialization of the (β − 1)-Grothendieck polynomial may have negative coefficients.
Our main objective in Section 5.2 is to study the polynomials R w (q, β) for a special class of permutations in the symmetric group S ∞ . Namely, in Section 5.2 we study some combinatorial properties of polynomials R ̟ λ,φ (q, β) for the five parameters family of vexillary permutations {̟ λ,φ } which have the shape λ := λ n,p,b = (p(n − i + 1) + b, i = 1, . . . , n + 1) and flag φ := φ k,r = (k + r(i − 1), i = 1, . . . , n + 1). This class of permutations is notable for many reasons, including that the specialized value of the Schubert polynomial S ̟ λ,φ (1) admits a nice product formula 11 , see Theorem 5.6. Moreover, we describe also some interesting connections of polynomials R ̟ λ,φ (q, β) with plane partitions, the Fuss-Catalan numbers 12 and Fuss-Narayana polynomials, k-triangulations and k-dissections of a convex polygon, as well as a connection with two families of ASM .
For example, let λ = (b n ) and φ = (k n ) be rectangular shape partitions, then the polynomial R ̟ λ,φ (q, β) defines a (q, β)-deformation of the number of (ordinary) plane partitions 13 sitting in the box b × k × n. It seems an interesting problem to find an algebra-geometric interpretation of polynomials R w (q, β) in the general case.
Question Let a and b be mutually prime positive integers. Does there exist a family of permutations w a,b ∈ S ab(a+b) such that the specialization x i = 1 ∀i of the Schubert polynomial S w a,b is equal o the rational Catalan number C a/b ? That is Many of the computations in Section 5.2 are based on the following determinantal formula for β-Grothendieck polynomials corresponding to grassmannian permutations, cf [59]. If w = σ λ is the grassmannian permutation with shape λ = (λ , . . . , λ n ) and a unique descent at position n, then 14 , where X n = (x i , x 1 , . . . , x n ), and for any set of variables X, 11 One can prove a product formula for the principal specialization S̟ λ,φ (xi := q i−1 , ∀i ≥ 1) of the corresponding Schubert polynomial. We don't need a such formula in the present paper. 12 We define the (generalized) Fuss-Catalan numbers to be F C can be described as follows 13 Let λ be a partition. An ordinary plane partition (plane partition for short)bounded by d and shape λ is a filling of the shape λ by the numbers from the set {0, 1, . . . , d} in such a way that the numbers along columns and rows are weakly decreasing.
A reverse plane partition bounded by d and shape λ is a filling of the shape λ by the numbers from the set {0, 1, . . . , d} in such a way that the numbers along columns and rows are weakly increasing.
14 the equality , has been proved independently in [70].
and h k (X) denotes the complete symmetric polynomial of degree k in the variables from the set X. .
In Section 5.3 we give a partial answer on the question 6.C8(d) by R.Stanley [90]. In particular, we relate the reduced polynomial corresponding to monomial x a 2 12 · · · x n−1,n an n−2 j=2 n k=j+2 x jk , a j ∈ Z ≥0 , ∀j, with the Ehrhart polynomial of the generalized Chan-Robbins-Yuen polytope, if a 2 = . . . = a n = m + 1, cf [66], with a t-deformation of the Kostant partition function of type A n−1 and the Ehrhart polynomials of some flow polytopes, cf [67]. In Section 5.4 we investigate certain specializations of the reduced polynomials corresponding to monomials of the form x m 1 12 · · · x mn n−1,n , m j ∈ Z ≥0 .∀j. First of all we observe that the corresponding specialized reduced polynomial appears to be a piece-wise polynomial function of parameters m = (m 1 , . . . , m n ) ∈ (R ≥0 ) n , denoted by P m . It is an interesting problem to compute the Laplas transform of that piece-wise polynomial function. In the present paper we compute the value of the function P m in the dominant chamber C n = (m 1 ≥ m 2 ≥ . . . ≥ m n ≥ 0), and give a combinatorial interpretation of the values of that function in points (n, m) and (n, m, k), n ≥ m ≥ k.
For the reader convenience, in Appendix I-V we collect some useful auxiliary information about the subjects we are treated in the present paper.
Almost all results in Section 5 state that some two specific sets have the same number of elements. Our proofs of these results are pure algebraic. It is an interesting problem to find bijective proofs of results from Section 5 which generalize and extend remarkable bijective proofs presented in [98], [85], [91], [67] to the cases of • the β-Grothendieck polynomials, • the (small) Schröder numbers, • k-dissections of a convex (n + k + 1)-gon, • special values of reduced polynomials. We are planning to treat and present these bijections in (a) separate publication(s).
We expect that the reduced polynomials corresponding to the higher-order powers of the Coxeter elements also admit an interesting combinatorial interpretation(s). Some preliminary results in this direction are discussed in Comments 5.8.
At the end of Introduction I want to add two remarks.
(a) After a suitable modification of the algebra 3HT n , see [52], and the case β = 0 in [47], one can compute the set of relations among the (additive) Dunkl elements (defined in Section 2, (2.1)). In the case β = 0 and q ij = q i δ j−i,1 , 1 ≤ i < j ≤ n, where δ a,b is the Kronecker delta symbol, the commutative algebra generated by additive Dunkl elements (2.3) appears to be "almost" isomorphic to the equivariant quantum cohomology ring of the flag variety Fl n , see [52] for details. Using the multiplicative version of Dunkl elements (3.14), one can extend the results from [52] to the case of equivariant quantum K-theory of the flag variety Fl n , see [47]. (b) As it was pointed out previously, one can define an analogue of the algebra 3T (0) n for any (oriented) matroid M n , and state a conjecture which connects the Hilbert polynomial of the algebra 3T (0) n (M n ) ab , t) and the chromatic polynomial of matroid M n . We expect that algebra 3T (β=1) n (M n ) ab is isomorphic to the Gelfand-Varchenko algebra associated with matroid M. It is an interesting problem to find a combinatorial meaning of the algebra 3T  [47].
My special thanks are to Professor Anders Buch for sending me the programs for computation of the β-Grothendieck and double β-Grothendieck polynomials. Many results and examples in the present paper have been checked by using these programs, and Professor Ole Warnaar (University of Queenslad) for a warm hospitality and a kind interest and fruitful discussions of some results from [47] concerning hypergeometric functions.
These notes represent an update version of Section 5 of my notes [47], and are based on my talks given at

Dunkl elements
Let F n be the free associative algebra over Z with the set of generators {u ij , 1 ≤ i, j ≤ n}. In the subsequent text we will distinguish the set of generators {u ii } 1≤i≤n from that {u ij } 1≤i =j≤n , and set x i := u ii , i = 1, . . . , n.
(Additive Dunkl elements) The (additive) Dunkl elements θ i , i = 1, . . . , n, in the algebra F n are defined to be We are interested in to find "natural relations" among the generators {u ij } 1≤i,j≤n such that the Dunkl elements (2.1) are pair-wise commute. One of the natural conditions which is the commonly accepted in the theory of integrable systems, is Therefore in order to ensure that the Dunkl elements form a pair-wise commuting family, it's natural to assume that the following conditions hold • (Unitarity) i.e. the elements u ij + u ji are central.
• ("Conservation laws") [ n k=1 x k , u ij ] = 0 f or all i, j, (2.5) i.e. the element E := n k=1 x k is central, Definition 2.2. (Dynamical six term relations algebra 6DT n ) We denote by 6DT n the quotient of the algebra F n by the two-sided ideal generated by relations (2.2) − (2.6).
Clearly, the Dunkl elements (2.1) generate a commutative subalgebra inside of the algebra 6DT n , and the sum x i belongs to the center of the algebra 6DT n . Remark Occasionally we will call the Dunkl elements of the form (2.1) by dynamical Dunkl elements to distinguish the latter from truncated Dunkl elements, corresponding to the case x i = 0, ∀i.

2.1
Some representations of the algebra 6DT n 2.1.1 Dynamical Dunkl elements and equivariant quantum cohomology [25]) Given a set q 1 , . . . , q n−1 of mutually commuting parameters, define and set q ij = q ji in the case i > j. Clearly, that if i < j < k, then q ij q jk = q ik . Let z 1 , . . . , z n be a set of (mutually commuting) variables. Denote by P n := Z[z 1 , . . . , z n ] the corresponding ring of polynomials. We consider the variable z i , i = 1, . . . , n, also as the operator acting on the ring of polynomials P n by multiplication on the variable z i .
Let s ij ∈ S n be the transposition that swaps the letters i and j and fixes the all other letters k = i, j. We consider the transposition s ij also as the operator which acts on the ring P n by interchanging z i and z j , and fixes all other variables. We denote by the divided difference operators corresponding to the transposition s ij and the simple transposition s i := s i,i+1 correspondingly. Finally we define operator (cf [25] ) The operators ∂ (ij) , 1 ≤ i < j ≤ n, satisfy (among other things) the following set of relations (cf [25]) Thus the elements {z i , i = 1, . . . , n} and {u ij , 1 ≤ i < j ≤ n} define a representation of the algebra DCY B n , and therefore the Dunkl elements form a pairwise commuting family of operators acting on the ring of polynomials Z[q 1 , . . . , q n−1 ][z 1 , . . . , z n ], cf [25]. This representation has been used in [25] to construct the small quantum cohomology ring of the complete flag variety of type A n−1 .
(II) Consider degenerate affine Hecke algebra H n generated by the central element h, the elements of the symmetric group S n , and the mutually commuting elements y 1 , . . . , y n , subject to relations where s i stand for the simple transposition that swaps only indices i and i + 1. For i < j, let s ij = s i · · · s j−1 s j s j−1 · · · s i denotes the permutation that swaps only indices i and j. It is an easy exercise to show that Finally, consider a set of mutually commuting parameters {p ij , 1 ≤ i = j ≤ n, p ij + p ji = 0}, subject to the constraints If parameters {p ij } are invertible, and satisfy relations then one can rewrite the above displayed relations in the following form: Therefore there exist parameters {q 1 , . . . , q n } such that 1 + β/p ij = q i /q j , 1 ≤ i < j ≤ n. In other words, p ij = β q j q j −q j , 1 ≤ i < j ≤ n. However in general, there are many other types of solutions, for example, solutions related to the Heaviside function 15 H(x), namely, p ij = H(x i − x j ), x i ∈ R, ∀i, and its discrete analogue, see Example (III) below. In the both cases β = −1; see also Comments 2.3 for other examples.
To continue presentation of Example (II), define elements u ij = p ij s ij , 1 ≤ i = j ≤ n.
Let's stress that the elements y i and θ j do not commute in the algebra H n , but the symmetric functions of y 1 , . . . , y n , i.e. the center of the algebra H n , do.
A few remarks in order. First of all, u 2 ij = p 2 ij are central elements. Secondly, in the case h = 0 and y i = 0, ∀i, the equality DET M n (u; x 1 , . . . , x n ) = u n describes the set of polynomial relations among the Dunkl-Gaudin elements (with the following choice of parameters p ij = (q i − q j ) −1 are taken). And our final remark is that according to [35], Section 8, the quotient ring is isomorphic to the quantum equivariant cohomology ring of the cotangent bundle T * Fl n of the complete flag variety of type A n−1 , namely, with the following choice of quantum parameters: On the other hand, in [52] we computed the so-called multiparameter deformation of the equivariant cohomology ring of the complete flag variety of type A n−1 .
A deformation defined in [52] depends on parameters {q ij , 1 ≤ i < j ≤ n} without any constraints are imposed. For the special choice of parameters the multiparameter deformation of the equivariant cohomology ring of the type A n−1 complete flag variety Fl n constructed in [52], is isomorphic to the ring H q n . 16 For the reader convenience we remind [26] a definition of the quantum elementary polynomial e q k (x1, . . . , xn). Let q := {qij } 1≤i<j≤n be a collection of "quantum parameters", then  Let us fix a set of independent parameters {q 1 , . . . , q n } and define new parameters We set deg(q ij ) = 2, deg(p ij ) = 1, deg(h) = 1.
The new parameters {q ij } 1≤i<j≤n , do not free anymore, but satisfy rather complicated algebraic relations. We display some of these relations soon, having in mind a question: is there some intrinsic meaning of the algebraic variety defined by the set of defining relations among the "quantum parameters" {q ij } ?
Let us denote by A n,h the quotient ring of the ring of polynomials Q[h][x ij , 1 ≤ i < j ≤ n] modulo the ideal generating by polynomials f (x ij ) such that the specialization x ij = q ij of a polynomial f (x ij ), namely f (q ij ), is equal to zero. The algebra A n,h has a natural filtration, and we denote by A n = grA n,h the corresponding associated graded algebra.
To describe (a part of) relations among the parameters {q ij } let us observe that parameters {p ij } and {q ij } are related by the following identity Using this identity we can find the following relations among parameters in question Finally, we come to a relation of degree 8 among the "quantum parameters" {q ij } There are also higher degree relations among the parameters {q ij } some of whose in degree 16 follow from the deformed Plücker relation between parameters {p ij }: However, we don't know how to describe the algebra A n,h generated by quantum parameters {q ij } 1≤i<j≤n even for n=4. The algebra A n = gr(A n,h ) is isomorphic to the quotient algebra of Q[x ij , 1 ≤ i < j ≤ n] modulo the ideal generated by the set of relations between "quantum parameters" which correspond to the Dunkl-Gaudin elements {θ i } 1≤i≤n , see Section 3.2 below for details. In this case the parameters {q ij } satisfy the following relations (q 2 ij q 2 jk + q 2 ij q 2 ik + q 2 jk q 2 ik = 2 q ij q ik q jk (q ij + q jk + q jk ) which correspond to the relations (2.8) in the special case h = 0. One can find a set of relations in degrees 6, 7 and 8, namely for a given pair-wise distinct integers 1 ≤ i, j, k, l ≤ n, one has • one relation in degree 6 • three relations in degree 7 • one relation in degree 8 However we don't know does the list of relations displayed above, contains the all independent relations among the elements {q ij } 1≤i<j≤n in degrees 6, 7 and 8, even for n = 4. In degrees ≥ 9 and n ≥ 5 some independent relations should appear. Notice that the parameters {p ij = h q j q i −q j , i < j} satisfy the so-called Gelfand-Varchenko relations, see e.g. [45] p ij p jk = p ik p ij + p jk p ik + h p ik , i < j < k, whereas parameters {p ij = 1 q i −q j , i < j} satisfy the so-called Arnold relations  Finally, if we set q i := exp(h z i ) and take the limit lim h→0 , as a result we obtain the Dunkl-Gaudin parameter q ij = 1 (z i −z j ) 2 .
(III) Consider the following representation of the degenerate affine Hecke algebra H n on the ring of polynomials P n = Q[x 1 , . . . , x n ]: • the symmetric group S n acts on P n by means of operators • y i acts on the ring P n by multiplication on the variable x i : In the subsequent discussion we will identify the operator of multiplication by the variable x i , namely the operator y i , with x i . This time define u ij = p ij (s i − 1), if i < j and set u ij = −u ji if i > j, where parameters {p ij } satisfy the same conditions as in the previous example. Then Comments 2.3. Let us list a few more representations of the dynamical classical Yang-Baxter relations.
• (Trigonometric Calogero-Moser representation) Let i < j, define • (Mixed representation) We set u ij = −u ji , if i > j. In all cases we define Dunkl elements to be θ i = j =i u ij . Note that operators satisfy the three term relations: r ij r jk = r ik r ij + r jk r ik , and r jk r ij = r ij r jk + r ik r jk , and thus satisfy the classical Yang-Baxter relations.

2.1.2
Dunkl-Uglov representation of degenerate affine Hecke algebra [94] ( Step functions and the Dunkl-Uglov representations of the degenerate affine Hecke algebras) For any two real numbers Lemma 2.4. The functions η ij satisfy the following relations where δ x,y denotes the Kronecker delta function.
To introduce the Dunkl-Uglov operators [94] we need a few more definitions and notation. To start with, denote by ∆ ± i the finite difference operators: . Let as before, {s ij , 1 ≤ i = j ≤ n, s ij = s ji }, denotes the set of transpositions in the symmetric group S n . Recall that s ij ( To simplify notation, set Lemma 2.5.
The operators {u ± iu , 1 ≤ i < j ≤ n} satisfy the following relations From now on we assume that x i ∈ Z, ∀i, that is, we will work with the restriction of the all operators defined at beginning of Example (2.1 (c)), to the subset Z n ⊂ R n . It is easy to see that under the assumptions x i ∈ Z, ∀i, we will have Moreover, using relations (2.13), (2.14) one can prove that • The operators {u ± ij , 1 ≤ i < j < k ≤ n, } and ∆ ± i , i = 1, . . . , n satisfy the dynamical classical Yang-Baxter relations • ( [94]) The operators {s i := s i,i+1 , 1 ≤ i < n, and ∆ ± j , 1 ≤ j ≤ n} give rise to two representations of the degenerate affine Hecke algebra H n . In particular, the Dunkl-Uglov operators are mutually commute: • [y ij , y ik + y jk ] = 0 = [y ij + y ik , y jk ], if i < j < k.
(ii) The elements z 1 , . . . , z n generate the free associative algebra F n .
A representation of the extended Kohno-Drinfeld algebra has been constructed in [35], namely one can take where q 1 , . . . , q n stands for a set of mutually commuting quantum parameters, and {T denotes the set of generators of the Yangian Y (gl n ), see e.g. [69]. A proof that the elements {z i } 1≤i≤n and {y ij } 1≤i =j≤n satisfy the extended Kohno-Drinfeld algebra relations is based on the following relations, see e.g. [35], Section 3 il , i, j, k, l = 1, . . . , n, s ∈ Z ≥0 .

"Compatible" Dunkl elements and Manin matrices
("Compatible" Dunkl elements, Manin matrices and algebras related with weighted complete graphs rK n ) Let us consider a collection of generators {u ij , 1 ≤ i, j ≤ n, α = 1, . . . , r}, subject to the following relations • either the unitarity (the case of sign "+"), or the symmetry relations (the case of sign " -") 18 : We define global 3-term relations algebra 3T n,r as " compatible product" of the local 3-term relations algebras. Namely, we require that the elements It is easy to check that our request is equivalent to a validity of the following sets of relations among the generators {u Now let us consider local Dunkl elements ij , j = 1, . . . , n, α = 1, . . . , r.
It follows from the local 3-term relations (⋆) that for a fixed α ∈ [1, r] the local Dunkl elements either mutually commute (the sign "+"), or pairwise anticommute (the sign " -"). Similarly, the global 3-term relations imply that the global Dunkl elements . . , n also either mutually commute (the case " + ") or pairwise anticommute (the case " -"). Now we are looking for a set of relations among the local Dunkl elements which is a consequence of the commutativity (anticommutativity) of the global Dunkl elements. It is quite j ] ± , and the commutativity (or anticommutativity) of the global Dunkl elements for all (λ 1 , . . . , λ r ) ∈ R r is equivalent to the following set of relations  should be either a Manin matrix (the case " + "), or its super analogue (the case " -"). Clearly enough that a similar construction can be applied to the algebras studied in Section 2, I-III.,and thus it produces some interesting examples of the Manin matrices.
It is an interesting problem to describe the algebra generated by the local Dunkl elements {θ and a commutative subalgebra generated by the global Dunkl elements inside the former. It is also an interesting question whether or not the coefficients C 1 , . . . , C n of the column characteristic polynomial Det col | Θ n − t I n |= n k=0 C k t n−k of the Manin matrix Θ n generate a commutative subalgebra ? For a definition of the column determinant of a matrix, see e.g. [16].
However a close look at this problem and the question posed needs an additional treatment and has been omitted from the content of the present paper.
Here we are looking for a "natural conditions" to be imposed on the set of generators {u α ij } 1≤α≤r 1≤i,j≤n in order to ensure that the local Dunkl elements satisfy the commutativity (or anticommutativity) relations: j ] ± = 0, f or all 1 ≤ i < j ≤ n and 1 ≤ α, β ≤ r. The "natural conditions" we have in mind are: if i, j, k, l are distinct and 1 ≤ α, β ≤ r.
Finally we define a multiple analogue of the three term relations algebra, denoted by 3T ± (rK n ), to be the quotient of the global 3-term relations algebra 3T ± n,r modulo the two-sided ideal generated by the left hand sides of relations (1.5), (1.6) and that of the following relations ij ] ± = 0, for all i = j, α = β. The outputs of this construction are • noncommutative quadratic algebra 3T (±)(rKn) generated by the elements {u  We expect that the subalgebra generated by local Dunkl elements in the algebra 3T + (rK n ) is closely related (isomorphic for r = 2) with the coinvariant algebra of the diagonal action of the symmetric group S n on the ring of polynomials Q[X  n }. The algebra (3T − (2K n )) (−) ) anti has been studied in [47], and [7]. In the present paper we state only our old conjecture. Hilb where for any algebra A we denote by A anti the quotient of algebra A by the two-sided ideal generated by the set of anticommutators {ab + ba | (a, b) ∈ A × A}.
According to observation of M. Haiman [37], the number 2 n (n + 1) n−2 is thought of as being equal to to the dimension of the space of triple coinvariants of the symmetric group S n .

Non-unitary dynamical classical Yang-Baxter algebra DCY B n
Let A n be the quotient of the algebra F n by the two-sided ideal generated by the relations (2.2), (2.5) and (2.6). Consider elements Clearly, if i < j, then where the elements w ijk , i < j, have been defined in Lemma 2.1, (2.3). Therefore the elements θ i andθ j commute in the algebra A n .
In the case when x i = 0 for all i = 1, . . . , n, the relations are well-known as the non-unitary classical Yang-Baxter relations. Note that for a given triple of pair-wise distinct (i, j, k) we have in fact 6 relations. These six relations imply that [θ i ,θ j ] = 0. However, in general, • (Dynamical classical Yang-Baxter algebra DCY B n ) In order to ensure the commutativity relations among the Dunkl elements

Definition 2.4.
Define dynamical non-unitary classical Yang-Baxter algebra DN U CY B n to be the quotient of the free associative algebra Q {x 1≤i≤n }, {u ij } 1≤i =j by the two-sided ideal generated by the following set of relations • (Zero curvature conditions)
• (Crossing relations) • (Twisted dynamical classical Yang-Baxter relations) It is easy to see that the twisted classical Yang-Baxter relations for a fixed triple of distinct indices i, j, k contain in fact 3 different relations whereas the nonunitary classical Yang-Baxter relations contain 6 different relations for a fixed triple of distinct indices i, j, k.
• Define dynamical classical Yang-Baxter algebra DCY B n to be the quotient of the algebra DN U CY B n by the two-sided ideal generated by the elements k =i,j [u ik , u ij + u ji ], f or all i = j.

•
Define classical Yang-Baxter algebra CY B n to be the quotient of the dynamical classical Yang-Baxter algebra DCY B n by the set of relations x i = 0 f or i = 1, · · · , n. Examples 2.1.
(a) Define Clearly, p ij + p ji = 1. Now define operators u ij = p ij s ij , and the truncated Dunkl operators to be θ i = j =i u ij , i = 1, . . . , n. All these operators act on the field of rational functions Q(z 1 , . . . , z n ); the operator s ij = s ji acts as the exchange operator, namely, Note that this time one has p 12 p 23 = p 13 p 12 + p 23 p 13 −p 13 .
It is easy to see that the operators {u ij , 1 ≤ i = j ≤ n} satisfy relations (3.1), Section 3, and therefore, satisfy the twisted classical Yang-Baxter relations (2.11). As a corollary we obtain that the truncated Dunkl operators {θ i , i = 1, . . . , n} are pair-wise commute. Now consider the Dunkl operator D i = ∂ z i + h θ i , i = 1, . . . , n, where h is a parameter. Clearly that [∂ z i + ∂ z j , u ij ] = 0, and therefore [D i , D j ] = 0 ∀i, j. It easy to see that In such a manner we come to the well-known representation of the degenerate affine Hecke algebra H n .

Definition 2.6.
Denote by 3QT n (β, h) an associative algebra generated over the ring Z[β, h] {q ij } 1≤i<j≤n by the set of generators {x 1 , . . . , x n } and that {u ij } 1≤i =j≤n } subject to the set of relations As before we define the (additive) Dunkl elements to be It is clearly seen from the defining relations listed in Definition 2.3 that for any triple of distinct indices (i, j, k) the elements {x i , x j , x k , u ji , u ik , u jk } satisfy the twisted dynamical Yang-Baxter relations, and thus the Dunkl elements {θ i } 1≤i≤n generate a commutative subalgebra in the algebra 3QT n (β, h).
where e (q+h) k (z 1 , . . . , z n ) denotes the multiparameter quantum elementary polynomial corresponding to the set of parameters It is not difficult to see that the unitarity and crossing conditions imply the following relations i , u kl ] = 0 are valid for all indices i = j, k = l. As a consequence of these relations one can deduce that the all symmetric polynomials e k (X n ) := e k (x 1 , . . . , x n ), k = 1, . . . , n belong to the center of the algebra 3QT n (q, h), and therefore one has [θ i , e k (X n )] = 0 for all i and k. Let us denote by QH(β, h) a commutative subalgebra in the algebra 3QT n (β, h) generated by the elementary symmetric polynomials {e k (X n )} 1≤k≤n and the Dunkl elements {θ i } 1≤i≤n . It is an interesting problem to give a geometric/cohomological interpretation of the commutative algebra QH(β, h). We don't know any geometric interpretation of that commutative algebra, except the special case [52] β = 0, h j = 1, ∀j, q ij := q i δ i+1,j . (2.18) Under assumptions (2.12), the algebra QH(0, 0) isomorphic to the equivariant quantum cohomology QH * T (Fl n ) of the complete flag variety Fl n . Examples 2.2. Let us list the relations among the Dunkl elements in the algebra 3QT n (β, h).
We define the Dunkl elements θ i , i = 1, . . . , n, by the formula (2.1). It is necessary to stress that the Dunkl elements {θ} 1≤i≤n do not commute in the algebra 3QL n (β, h) but satisfy a noncommutative analogue of the relations displayed in Theorem 2.3. Namely, one needs to replace the both elementary polynomials e k (Z n ) and the quantum multiparameter elementary polynomials e (q) k (Z n ) by its noncommutative versions. Recall that the noncommutative elementary polynomial e k (Z n ) is equal to

Project 2.2. (Noncommutative universal Schubert polynomials)
Let w ∈ S n be a permutation and S w (Z n ) be the corresponding Schubert polynomial.
(1) There exists a (noncommutative) polynomial Sh w ({u ij } 1≤i<j≤n ) with non-negative integer coefficients such that the following identity 3) Let r ∈ Z ≥2 and N = n 1 + · · · n r ,, n j ∈ Z ≥1 , ∀j, be a composition of N , and set in the algebra 3QT n (β, h), by the use of the degree 1,. . . ,n r relations among the former. As a result one obtains a set consisting of N r−1 relations among the N r−1 elements Give a geometric interpretation of the commutative subalgebra QH n 1 ,...,nr (β, h) ⊂ 3QT n (β, h) generated by the set of elements θ (N ) j,k j , 1 ≤ k j ≤ n j , j = 1, . . . , r − 1.

Dunkl and Knizhnik-Zamolodchikov elements
• Assume that ∀i, x i = 0, and generators {u ij , 1 ≤ i < j ≤ n} satisfy the locality conditions (2.2) and the classical Yang-Baxter relations Let y, z, t 1 , . . . , t n be parameters, consider the rational function .
• Now assume that a set of generators {c ij , 1 ≤ i = j ≤ n} satisfy the locality and symmetry (i.e. c ij = c ji ) conditions, and the Kohno-Drinfeld relations: Let y, z, t 1 , . . . , t n be parameters, consider the rational function .

Dunkl and Gaudin operators
(a) ( Rational Dunkl operators) Consider the quotient of the algebra DCY B n , see Definition 2.2, by the two-sided ideal generated by elements Clearly the Dunkl elements (2.1) mutually commute. Now let us consider the so-called Calogero-Moser representation of the algebra DCY B n on the ring of polynomials R n := R[z 1 , . . . , z n ] given by The symmetric group S n acts on the ring R n by means of transpositions s ij ∈ S n : In the Calogero-Moser representation the Dunkl elements θ i becomes the rational Dunkl operators [21], see Definition 1.1. Moreover, one has [x k , u ij ] = 0, if k = i, j, and The Dunkl-Gaudin representation of the algebra DCY B n is defined on the field of rational functions K n := R(q 1 , . . . , q n ) and given by but this time we assume that w(q i ) = q i , ∀i ∈ [1, n] and for all w ∈ S n . In the Dunkl-Gaudin representation the Dunkl elements becomes the rational Gaudin operators, see e.g. [71]. Moreover, one has [x k , u ij ] = 0, if k = i, j, and

Comments 2.4.
It is easy to check that if f ∈ R[z 1 , . . . , z n ], then the following commutation relations are true Using these relations it easy to check that in the both cases (a) and (b) the elementary symmetric polynomials e k (x 1 , . . . , x n ) commute with the all generators {u ij } 1≤i,j≤n , and therefore commute with the all Dunkl elements {θ i } 1≤i≤n . Let us stress that [θ i , x k ] = 0 for all 1 ≤ i, k ≤ n.
Describe a commutative algebra generated by the Dunkl elements {θ i } 1≤i≤n and the ele- 2.3.6 Representation of the algebra 3T n on the free algebra Z t 1 , . . . , t n .
Let F n = Z t 1 , . . . , t n be free associative algebra over the ring of integers Z, equipped with the action of the symmetric group S n : Define the action of u ij ∈ 3T n on the set of generators of the algebra F n as follows The action of generator u ij on the whole algebra F n is defined by linearity and the twisted Leibniz rule: It is easy to see from (2.15) that (2.20) Now let us consider operator Let F • n be the quotient of the free algebra F n by the two-sided ideal generated by elements One can also define a representation of the algebra 3T n denotes the quotient of the algebra F n by the two-sided ideal generated by elements n , but the elements u i,j u i,k u j,k u i,j , 1 ≤ i < j < k ≤ n, from the kernel of the Calogero-Moser representation, act trivially both on the algebras F whereas the algebra F • n is not Koszul for n ≥ 3, and

Fulton universal ring, multiparameter quantum cohomology and F KT L
(The Fulton universal ring [31], multiparameter quantum cohomology of flag varieties [26]

and the full Kostant-Toda lattice [30])
Let X n = (x 1 , . . . , x n ) be be a set of variables, and Let t be an auxiliary variable, denote by M = (m ij ) 1≤i,j≤n the matrix of size n by n with the following elements: Let P n (X n , t) = det|M |.
Definition 2.8. The Fulton universal ring R n−1 is defined to be the quotient 19 Lemma 2.9. Let P n (X n , t) = n k=0 c k (n)t n−k , c 0 (n) = 1. Then c k (n) := c k (n; X n , g (n) ) = 1≤i 1 <i 2 <...<is<n where in the summation we assume additionally that the sets [i a , i a + j a ] := {i a , i a + 1, . . . , i a + j a }, a = 1, . . . , s, are pairwise disjoint.

It is clear that
One can easily see that the coefficients c k (n) and g m [k] satisfy the following recurrence relations [31]: On the other hand, let {q ij } 1≤i<j≤n be a set of (quantum) parameters, and e (q) k (X n ) be the multiparameter quantum elementary polynomial introduced in [26]. We are interested in to describe a set of relations between the parameters {g i [j]} i≥1,j≥1 i+j≤n and the quantum parameters To start with, let us recall the recurrence relations among the quantum elementary polynomials, cf [76]. To do so, consider the generating function Parameters {g a [b]} can be expressed polynomially in terms of quantum parameters {q ij } and variables x 1 , . . . , x n , in a such way that Moreover, • The quantum parameters {q ij } can be presented as rational functions in terms of variables x 1 , . . . , x n and polynomially in terms of parameters {g a [b]} such that the equality c k (n) = e (q) k (X n ) holds for all k, n.
In other words, the transformation defines a "birational transformation" between the algebra Z[g (n) ][X n ]/ P n (X n , t) − t n and multiparameter quantum deformation of the algebra H * (Fl n , Z).

Example 2.2.
Clearly, g n−1 [1] = n−1 j=1 q j,n , n ≥ 2 and g n−2 [2] = n−2 j=1 q jn (x n−1 − x j ), n ≥ 3. Moreover  (b), for the definition of the F KT L and its basic properties. In the present paper we just want to point out on a connection of the Fulton universal ring and hence the multiparameter deformation of the cohomology ring of complete flag varieties, and polynomial integral of motion of the FKTL. Namely, Polynomials c k (n; X n , g (n) ) defined by (2.17) coincide with the polynomial integrals of motion of the FKTL.
It seems an interesting task to clarify a meaning of the F KT L rational integrals of motion in the context of the universal Schubert Calculus [31] and the algebra 3HT n (0), as well as any meaning of universal Schubert or Grothendieck polynomials in the context of the Toda or full Kostant-Toda lattices.

Algebra 3HT n
Consider the twisted classical Yang-Baxter relation Having in mind applications of the Dunkl elements to Combinatorics and Algebraic Geometry, we split the above relation into two relations and impose the following unitarity constraints where β is a central element. Summarizing, we come to the following definition.
Define algebra 3T n (β) to be the quotient of the free associative algebra It is clear that the elements {u ij , u jk , u ik , 1 ≤ i < j < k ≤ n} satisfy the classical Yang-Baxter relations, and therefore, the elements {θ i := j =i u ij , 1 = 1, . . . , n} form a mutually commuting set of elements in the algebra 3T n (β).
(1) The elements {q ij , 1 ≤ i < j ≤ n} satisfy the Kohno-Drinfeld relations ( known also as the horizontal four term relations) Then (3) (Deviation from the Yang-Baxter and Coxeter relations) Finally we introduce the "Hecke quotient" of the algebra 3T n (β), denoted by 3HT n (β).
Definition 3.3. Define algebra 3HT n (β) to be the quotient of the algebra 3T n (β) by the set of relations q ij q kl = q kl q ij , f or all i, j, k, l.
In other words we assume that the all elements {q ij , 1 ≤ i < j ≤ n} are central in the algebra 3T n (β). From Lemma 3.1 follows immediately that in the algebra 3HT n (β) the elements {u ij } satisfy the multiplicative (or quantum) Yang-Baxter relations be a set of mutually commuting elements.

Definition 3.4.
Modified 3-term relation algebra 3M T n (β, ψ) is an associative algebra over the ring of polynomials Z[β, q ij , ψ ij ] with the set of generators {u ij , 1 ≤ i, j ≤ n} subject to the set of relations It is easy to see that in the algebra 3M T n (β, ψ) the generators {u ij } satisfy the modified Coxeter and modified quantum Yang-Baxter relations, namely It is still possible to describe relations among the additive Dunkl elements [47], cf [50]. However we don't know any geometric interpretation of the commutative algebra obtained. It is not unlikely that this commutative subalgebra is a common generalization of the small quantum cohomology and elliptic cohomology (remains to be defined !) of complete flag varieties.
The algebra 3M T n (β = 0, ψ) has an elliptic representation [47], [50]. Namely, where {λ i , i = 1, . . . , n} is a set of parameters (e.g. complex numbers), and {z 1 , . . . , z n } is a set of variables; s ij , i < j, denotes the transposition that swaps i on j and fixes all other variables; denotes the Kronecker sigma function; ℘(z) denotes the Weierstrass P -function. ◮ ◮ ("Multiplicative" version of the elliptic representation) Let q be parameter. In this place we will use the same symbol θ(x) to denote the "multiplicative" version of the Riemann theta function Let us state some well-known properties of the Riemann theta function : One can easily check that after the change of variables the functional equation for the Riemann theta function θ(x) takes the following form where σ λ (z) := θ(z/λ) θ(z) θ(λ −1 denotes the Kronecker sigma function. Therefor, the operators where s ij denotes the exchange operator which swaps the variables z i and z j , namely s ij (z i ) = z j , s ij (z j ) = z i , s ij (z k ) = z k , ∀k = i, j, and s ij acts trivially on dynamical parameters λ i , namely, s ij (λ k ) = λ k , ∀k, give rise to a representation of the algebra 3M T n (0, ψ). ◭ ◭ The 3-term relations among the elements {u ij } are consequence (in fact equivalent) to the famous Jacobi-Riemann 3-term relation of degree 4 among the theta function θ(z), see e.g. [97], p.451, Example 5. In several cases, see Introduction, relations (A) and (B), identities among the Riemann theta functions can be rewritten in terms of the elliptic Kronecker sigma functions and turn out to be a consequence of certain relations in the algebra 3M T n (0, ψ) for some integer n, and vice versa 20 .
The algebra 3HT n (β) is the quotient of algebra 3M T n (β, ψ) by the two-sided ideal generated by the elements {ψ ij }. Therefore the elements {u ij } of the algebra 3HT n (β) satisfy the quantum Yang-Baxter relations u ij u ik u jk = u jk u ik u ij , i < j < k, and as a consequence, the multiplicative Dunkl elements generate a commutative subalgebra in the algebra 3HT n (β), see Section 3.1. We emphasize that the Dunkl elements Θ j , j = 1, . . . , n, do not pairwise commute in the algebra 3M T n (β, ψ), if ψ ij = 0 for some i = j. One way to construct a multiplicative analog of additive Dunkl elements θ i := j =i u ij is to add a new set of mutually commuting generators denoted by {ρ ij , ρ ij + ρ ji = 0, 1 ≤ i = j ≤ n} subject to crossing relations • ρ ij commutes with β, q kl and ψ k,l for all i, j, k, l, Under these assumptions one can check that elements In the case of elliptic representation defined above, one can take where µ ∈ C * is a parameter. This solution to the quantum Yang-Baxter equation has been discovered in [86]. It can be seen as an operator form of the famous (finite dimensional) solution to QY BE due to A. Belavin and V. Drinfeld [5]. One can go one step more and add to the algebra in question new set of generators corresponding to the shift operators T i,q : z i −→ q z i , cf [24]. In this case one can define multiplicative Dunkl elements which are closely related with the elliptic Ruijsenaars-Schneider-Macdonald operators. 20 It is commonly believed that any identity between the Riemann theta functions is a consequence of the Jacobi-Riemann three term relations among the former. However we do not expect that the all hypergeometric type identities among the Riemann theta functions can be obtained from certain relations in the algebra 3M Tn(0, ψ) after applying the elliptic representation of the latter.

Multiplicative Dunkl elements
Since the elements u ij , u ik and u jk , i < j < k, satisfy the classical and quantum Yang-Baxter relations (3.1) and(3.2), one can define a multiplicative analogue denoted by Θ i , 1 ≤ i ≤ n, of the Dunkl elements θ i . Namely, to start with, we define elements We consider h ij (t) as an element of the algebra 3HT n : , where we assume that the all parameters {q ij , t, x, y, . . .} are central in the algebra 3HT n .
and therefore the elements (4) Define multiplicative Dunkl elements (in the algebra 3HT n ) as follows Then the multiplicative Dunkl elements pair-wise commute.
Here for a subset I ⊂ [1, n] we use notation Θ I = a∈I Θ a , Our main result of this Section is a description of relations among the multiplicative Dunkl elements. In the algebra 3HT n (β) the following relations hold true Here n k q denotes the q-Gaussian polynomial. Corollary 3.1.
Assume that q ij = 0 for all 1 ≤ i < j ≤ n. Then the all elements {u ij } are invertible and Then we have (1) (Relationship among Θ j and Φ j ) (2) The elements {Φ i , 1 ≤ i ≤ n, } generate a commutative subalgebra in the algebra 3HT n .
(3) For each k = 1, . . . , n, the following relation in the algebra 3HT n among the elements In fact the element Φ i admits the following "reduced expression" (i.e. one with the minimal number of terms involved) which is useful for proofs and applications Let us explain notations. For any (totally) ordered set I = (i 1 < i 2 < . . . < i k ) we denote by I + the set I with the opposite order, i.e.
, then we set I c := [1, n] \ I. For any (totally) ordered set I we denote by − −− → i∈I the ordered product according to the order of the set I.
Note that the total number of terms in the RHS of (3.4) is equal to i(n − i).
Finally, from the "reduced expression" (3.4) for the element Φ i one can see that Therefore the identity is true in the algebra 3HT n for any set of parameters {q ij }.

Comments 3.2.
In fact from our proof of Theorem 3.1 we can deduce more general statement, namely, consider integers m and k such that 1 ≤ k ≤ m ≤ n. Then where , by definition, for two sets A = (i 1 , . . . , i r ) and B = (j 1 , . . . , j r ) the symbol u A,B is equal to the (ordered) product r a=1 u ia,ja . Moreover, the elements of the sets A and B have to satisfy the following conditions: • for each a = 1, . . . , r one has 1 ≤ i a ≤ m < j a ≤ n, and k ≤ r ≤ k(n − k). Even more, if r = k, then sets A and B have to satisfy the following additional conditions: , and the elements of the set A are pair-wise distinct.
In the case β = 0 and r = k, i.e. in the case of additive (truncated) Dunkl elements, the above statement, also known as the quantum Pieri formula, has been stated as Conjecture in [26], and has been proved later in [76].

Corollary 3.2. ([51])
In the case when β = 0 and q ij = q i δ j−i,1 , the algebra over Z[q 1 , . . . , q n−1 ] generated by the multiplicative Dunkl elements {Θ i and Θ −1 i , 1 ≤ i ≤ n} is canonically isomorphic to the quantum K-theory of the complete flag variety Fl n of type A n−1 .
It is still an open problem to describe explicitly the set of monomials {u A,B } which appear in the RHS of (3.5) when r > k.

Truncated Gaudin operators
Let {p ij 1 ≤ i = j ≤ n} be a set of mutually commuting parameters. We assume that parameters {p ij } 1≤i<j≤n are invertible and satisfy the Arnold relations For example one can take Definition 3.5. Truncated (rational) Gaudin operator corresponding to the set of parameters where s ij denotes the exchange operator which switches variables x i and x j , and fixes parameters {p ij }.
We consider the Gaudin operator G i as an element of the group ring ij }][S n ], i = 1, . . . , n, by Gaudin element and denoted it by θ It is easy to see that the elements u ij := p −1 ij s ij , 1 ≤ i = j ≤ n, define a representation of the algebra 3HT n (β) with parameters β = 0 and q ij = u 2 ij = p 2 ij . Therefore one can consider the (truncated) Gaudin elements as a special case of the (truncated) Dunkl elements. Now one can rewrite the relations among the Dunkl elements, as well as the quantum Pieri formula [26] , [76], in terms of the Gaudin elements.
The key observation which allows to rewrite the quantum Pieri formula as a certain relation among the Gaudin elements, is the following one: parameters {p −1 ij } satisfy the P lücker relations To describe relations among the Gaudin elements θ (n) i , i = 1, . . . , n, we need a bit of notation. Let {p ij } be a set of invertible parameters as before. i a < j a , a = 1, . . . , r. Define polynomials in the variables h = (h 1 , . . . , h n ) and summation runs over subsets (Relations among the Gaudin elements, [47], cf [71]) Under the assumption that elements {p ij , 1 ≤ i < j ≤ n} are invertible, mutually commute and satisfy the Arnold relations, one has where d 2 , . . . , d n denote the Jucys-Murphy elements in the group ring Z[S n ] of the symmetric group S n , see Comments 3.1 for a definition of the Jucys-Murphy elements.
It is well-known that the elementary symmetric polynomials e r (d 2 , . . . , d n ) := C r , r = 1, . . . , n − 1, generate the center of the group ring Z[p ±1 ij ][S n ], whereas the Gaudin elements {θ It follows from [71] that the relations (3.6) together with relations are the defining relations for the algebra B λ (p ij ).
Let us remark that in the definition of the Gaudin elements we can use any set of mutually commuting, invertible elements {p ij } which satisfies the Arnold conditions. For example, we can take where as before, d j denotes the Jucys-Murphy element in the group ring Z[S n ] of the symmetric group S n . Basically from relations (2.15) one can deduce the relations among the Jucys-Murphy elements d 2 , . . . , d n after plugging in (3.6) the values p ij := q j−2 (1−q) 1−q j−i and passing to the limit q → 0. However the real computations are rather involved.
Finally we note that the multiplicative Dunkl / Gaudin elements {Θ i , 1, . . . , n} also generate a maximal commutative subalgebra in the group ring Z[p ±1 ij ][S n ]. Some relations among the elements {Θ l } follow from Theorem 3.2, but we don't know an analogue of relations (3.6) for the multiplicative Gaudin elements, but see [71].
Let A = (a i,j ) be a 2m × 2m skew-symmetric matrix. The Pfaffian and Hafnian of A are defined correspondingly by the equations where S 2m is the symmetric group and sgn(σ) is the signature of a permutation σ ∈ S 2m , see e.g. http://en.wikipedia.org/wiki/Pfaffian. Now let n be a positive integer, and {p ij , 1 ≤ i = j ≤ n, p ij + p ji = 0} be a set of skew-symmetric, invertible and mutually commuting elements. We set p ii = 0 for all i, and q := {p 2 ij } 1≤i<j≤n . Now let us assume that the elements {p ij } 1≤i<j≤n satisfy the Plüker relations for the elements (a) Let n be an even positive integer. Let us define A n (p ij ) := (p ij ) 1≤i,j≤n to be the n × n skew-symmetric matrix corresponding to the family {p ij } 1≤i<j≤n .
On the other hand, if one assumes that a set of skew symmetric parameters {r ij } 1≤i<j≤n , r ij + r ji = 0, satisfies the "standard" Plüker relations, namely

Shifted Dunkl elements d i and D i
As it was stated in Corollary 3.2, the truncated additive and multiplicative Dunkl elements in the algebra 3HT n (0) generate over the ring of polynomials Z[q 1 , . . . , q n−1 ] correspondingly the quantum cohomology and quantum K − theory rings of the full flag variety Fl n . In order to describe the corresponding equivariant theories, we will introduce the shifted additive and multiplicative Dunkl elements. To start with we need at first to introduce an extension of the algebra 3HT n (β).
Now we set as before h ij := h ij (t) = 1 + t u ij . Definition 3.7.
• Define shifted additive Dunkl elements to be • Define shifted multiplicative Dunkl elements to be Now we stated an analogue of Theorem 3.1. for shifted multiplicative Dunkl elements. As a preliminary step, for any subset I ⊂ [1, n] let us set D I = a∈I D a . It is clear that In the algebra 3HT n (β, h) the following relations hold true In particular, if β = 0, we will have Corollary 3.3. In the algebra 3HT n (0, h) the following relations hold (3.9) , then the subalgebra generated by multiplicative Dunkl elements D i , i = 1, . . . , n, in the algebra 3HT n (0, h = 1) (and t = 1), | is isomorphic to the equivariant quantum K-theory of the complete flag variety Fl n .
Our proof is based on induction on k and the following relations in the algebra 3HT n (β, h) if i < j < k, and we set h ij := h ij (1). These relations allow to reduce the left hand side of the relations listed in Theorem 3.3 to the case when z i = 0, h i = 0, ∀i. Under these assumptions one needs to proof the following relations in the algebra 3HT n (β), see Theorem 3.1, In the case β = 0 the identity (3.9) has been proved in [51] One of the main steps in our proof of Theorem 3.1. is the following explicit formula for the elements D I .

Lemma 3.4. One has
Note that if a < b, then h ba = 1 + βt − u ab . Here we have used the symbol For example, let us take n = 6 and I = (1, 3, 5), then Let us stress that the element D I ∈ 3HT n (β) is a linear combination of square free monomials and therefore, a computation of the left hand side of the equality stated in Theorem 3.3 can be performed in the "classical case" that is in the case q ij = 0, ∀i < j. This case corresponds to the computation of the classical equivariant cohomology of the type A n−1 complete flag variety Fl n , if h = 1.
A proof of the β = 0 case given in [51], Theorem 1, can be immediately extended to the case β = 0.
(1) Show that (2) ((β, h)-Stirling polynomials of the second type) Define polynomials S n.k (β, h) as follows Show that 4 Algebra 3T Let's consider the set R n := {(i, j) ∈ Z × Z | 1 ≤ i < j ≤ n} as the set of edges of the complete graph K n on n labeled vertices v 1 , . . . , v n . Any subset S ⊂ R n is the set of edges of a unique subgraph Γ := Γ S of the complete graph K n .

Definition 4.1. (Graph and nil-graph subalgebras)
The graph subalgebra 3T n (Γ) (resp. nil-graph subalgebra 3T (0) n (Γ)) corresponding to a subgraph Γ ⊂ K n of the complete graph K n , is defined to be the subalgebra in the algebra 3T n (resp.3T In subsequent Subsections 4.1.1 and 4.1.2 we will study some examples of graph subalgebras corresponding to the complete multipartite graphs, cycle graphs and linear graphs.   (A) (cf [4]) The subalgebra N C n is canonically isomorphic to the NilCoxeter algebra N C n . In particular, Hilb( N C n , t) = [n] t !.
(B) The subalgebra AN C n has finite dimension and its Hilbert polynomial is equal to is generated by the following elements: Note that deg f n which does not contain the generator u 1,n , can be written as a linear combination of degree k(n − k) monomials in the algebra AN C n , each contains the generator u 1,n at least once. By this means we obtain a set of all extra relations (i.e. additional to those in the algebra N C n ) in the algebra AN C n . Moreover, each monomial M in all linear combinations mentioned above, appears with coefficient Let B W be the corresponding Nichols-Woronowicz algebra, see e.g. [4]. Follow [4], denote by N C W the subalgebra in B W generated by the elements [α s ] ∈ B W corresponding to simple roots s ∈ S. Denote by AN W C W the subalgebra in B W generated by N C W and the element [a θ ], where [a θ ] stands for the element in B W corresponding to the highest root θ for W. In other words, AN W C W is the image of the algebra AN C W under the natural map BE(W ) −→ B W , see e.g. [4], [49]. It follows from [4], Section 6, that where denotes the Poincaré polynomial corresponding to the affine Weyl group W af f , see [12], p.245; a i := (2ρ, α ∨ i ), 1 ≤ i ≤ l, denote the coefficients of the decomposition of the sum of positive roots 2ρ in terms of the simple roots α i .
is a symmetric (and unimodal ?) polynomial with non-negative integer coefficients.

Parabolic 3-term relations algebras and partial flag varieties
In fact one can construct an analogue of the algebra 3HT n and a commutative subalgebra inside it, for any graph Γ = (V, E) on n vertices, possibly with loops and multiple edges, [47]. We denote this algebra by 3T n (Γ), and denote by 3T (0) n (Γ) its nil-quotient, which may be considered as a "classical limit of the algebra 3T n (Γ)". The case of the complete graph Γ = K n reproduces the results of the present paper and those of [47], i.e. the case of the full flag variety Fl n . The case of the complete multipartite graph Γ = K n 1 ,...,nr reproduces the analogue of results stated in the present paper for the full flag variety Fl n , to the case of the partial flag variety F n 1 ,...,nr , see [47] for details. We expect that in the case of the complete graph with all edges having the same multiplicity m, denoted by Γ = K (m) n , or mK n in the present paper, the commutative subalgebra generated by the Dunkl elements in the algebra 3T It is not difficult to see that 21 4, 6, 3).
Here for any algebra A we denote by A ab its abelianization 22 .
To continue exposition, let us take m ≤ n, and consider the complete multipartite graph K n,m which corresponds to the grassman variety Gr(n, m + n.) One can show where n k := S(n, k) denotes the Stirling numbers of the second kind, that is the number of ways to partition a set of n labeled objects into k nonempty unlabeled subsets, and for any graph Γ, T utte(Γ, x, y) denotes the Tutte polynomial 23 corresponding to graph Γ.
where B (k) n denotes the poly-Bernoulli number introduced by M. Kaneko [42]. For the reader's convenient, we recall below a definition of poly-Bernoulli numbers. To start with, let k be an integer, consider the formal power series If k ≥ 1, Li k (z) is the k-th polylogarithm, and if k ≤ 0, then Li k (z) is a rational function. Clearly Li 1 (z) = −ln(1 − z). Now define poly-Bernoulli numbers through the generating function Note that a combinatorial formula for the numbers B .
Let us recall, see Section 2, footnote 16, that for a commutative ring R and a polynomial p(t) = s j=1 g j t j ∈ R[t], we denote by p(t) the ideal in the ring R generated by the coefficients g 1 , . . . , g s .
These examples are illustrative of the similar results valid for the general complete multipartite graphs K n 1 ,...,nr , i.e. for the partial flag varieties [47].
To state our results for partial flag varieties we need a bit of notation. Let N := n 1 + . . . + n r , n j > 0, ∀j, be a composition of size N. We set N j := n 1 + · · · + n j , j = 1, . . . , r, and N 0 = 0, Now, consider the commutative subalgebra in the algebra 3T The commutative subalgebra generated by the elements {c Recall that a point F of the partial flag variety Fl n 1 ,...,nr , n 1 + · · · + n r = N, is a sequence of embedded subspaces By definition, the fiber of the vector bundle ξ i over a point F ∈ Fl n 1 ,...,nr is the n i -dimensional vector space F i /F i−1 .
The above examples show that the Hilbert polynomial Hilb(3T 0 n (G) ab , t) appears to be a certain specialization of the Tutte polynomial of the corresponding graph G. Instead of using the Hilbert polynomial of the algebra 3T 0 n (G) ab one can consider the graded Betti numbers polynomial Betti(3T 0 n (G) ab , x, y). For example, (1 + jt), where we set |k| := k 1 + . . . + k r .

Indeed, one can show 24
Proposition 4.2.
If r ∈ Z ≥1 , then where by definition (t) m := m−1 j=1 (t − j), (t) 0 = 1, (t) m = 0, if m < 0. Finally we describe explicitly the exponential generating function for the Tutte polynomials of the weighted complete multipartite graphs. We refer the reader to [68] for a definition and a list of basic properties of the Tutte polynomial of a graph. The ℓ-weighted complete multipartite graph K (ℓ) n 1 ,...,nr is a graph with the set of vertices equals to the disjoint union r j=1 S i of the sets S 1 , . . . , S r , and the set of edges {(α i , β j ), α i ∈ S i , β j ∈ S j } 1≤i<j≤r of multiplicity ℓ ij each edge 9α , β j ). Let us stress that to abuse of notation the complete unipartite graph K (n) consists of n disjoint points with the Tutte polynomial equals to 1 for all n ≥ 1, whereas the complete graph Kn is equal to the complete multipartite graph K (1 n ) . Let m = (m ij , 1 ≤ i < j ≤ n) be a collection of non-negative integers. Define generalized Tutte polynomial T n (m, x, y) as follows : .
n is a subgraph of the weighted complete graph K (ℓ) 1 n , then the Tutte polynomial of graph Γ multiplied by (x − 1) κ(Γ) is equal to the following specialization   Explicitly, . Further specialization q ij −→ 0, if edge(i, j) / ∈ Γ allows to compute the Tutte polynomial for any graph. (1) Assume that ℓ ij = ℓ for all 1 ≤ i < j ≤ r. Based on the above formula for the exponential generating function for the Tutte polynomials of the complete multipartite graphs K n 1 ,...,nr , deduce the following well-known formula where N := n 1 + · · · + n r . It is well-known that the number T utte(Γ, 1, 1) is equal to the number of spanning trees of a connected graph Γ.

Exercises 4.2.
Let K n 1 ,...,nr be complete multipartite graph, N := n 1 + · · · + n r . Show that 25 Hilb(3T N (K n 1 ,...,nr ), t) = r j=1 In this Section we introduce an analogue of the algebra 3T n (β) for the classical root systems. ( x ik y jk = y jk y ij + y ij x ik + β y ij , y ik x jk = x jk y ij + y ij y ik + β y ij , if 1 ≤ i < j < k ≤ n, (6) (Four term relations) The associative classical Yang-Baxter algebra ACY B(B n ) of type B n is the special case β = 0 of the algebra ACY B(B n ).

Comments 4.2.
• In the case β = 0 the algebra ACY B(B n ) has a rational representation • In the case β = 1 the algebra ACY B(B n ) has a "trigonometric" representation Definition 4.7. The bracket algebra E(B n ) of type B n is an associative algebra with the set of generators {x ij , y ij , z i , 1 ≤ i = j ≤ n} subject to the set of relations (1) − (6) listed in Definition 4.6, and the additional relations (5a) x jk x ij = x ij x ik + x ik x jk − β x ik , y jk x ij = x ij y ik + y ik y jk − β y ik , y jk x ik = y ij y jk + x ik y ij + β y ij , x jk y ik = y ij x jk + y ik y ij + β y ij , 25 It should be remembered that to abuse of notation, the complete graph Kn, by definition, is equal to the complete multipartite graph K ((1, . . . , 1) n ), whereas the graph K (n) is a collection of n distinct points.
Let us introduce the following Coxeter type elements: x a,a+1 z n ∈ E(B n ), and h Dn := n−1 a=1 x a,a+1 y n−1,n ∈ E(D n ). Let us bring the element h Bn (resp. h Dn ) to the reduced form in the algebra E(B n ) that is, let us consecutively apply the defining relations (1)−(6), (5a, 6a) to the element h Bn (resp. apply to h Dn the defining relations for algebra E(D n ) ) in any order until unable to do so. Denote the the resulting (noncommutative) polynomial by P Bn (x ij , y ij , z) (resp. P Dn (x ij , y ij )). In principal, this polynomial itself can depend on the order in which the relations (1) − (6), (5a, 6a) are applied. (1) Apart from applying the commutativity relations (1)−(4) , the polynomial P Bn (x ij , y ij , z) (resp. P Dn (x ij , y ij )) does not depend on the order in which the defining relations have been applied.
(2) Define polynomial P Bn (s, r, t) (resp. P Dn (s, r)) to be the the image of that P Bn (x ij , y ij , z) (resp. P Dn (x ij , y ij )) under the specialization

Super analogue of 6-term relations and classical Yang-Baxter algebras
4.2.1 Six term relations algebra 6T n , its quadratic dual (6T n ) ! , and algebra 6HT n Definition 4.9.
We denote by CY B n , named by classical Yang-Baxter algebra, an associative algebra over Q generated by elements {r ij , 1 ≤ i = j ≤ n} subject to relations 1) and 3).
Note that the algebra 6T n is given by n 2 generators and n 3 + 3 n 4 quadratic relations. Definition 4.10.
Define Dunkl elements in the algebra 6T n to be It easy to see that the Dunkl elements {θ i } 1≤i≤n generate a commutative subalgebra in the algebra 6T n . Let A = U (sl (2)) be the universal enveloping algebra of the Lie algebra sl (2). Recall that the algebra sl (2) It is not hard to see that • there are three rational solutions: and r 3 (u, v) := −r 2 (v, u).
• there is a trigonometric solution Notice that the Dunkl element θ j := a =j r trig (u a , u j ) corresponds to the truncated (or level 0) trigonometric Knizhnik-Zamolodchikov operator.
In fact, the "sl n -Casimir element" Ω = 1  n the quotient of the algebra 6T n by the (two-sided) ideal generated by the set of elements {r 2 i,j , 1 ≤ i < j ≤ n}. More generally, let {β, q ij , 1 ≤ i < j ≤ n} be a set of parameters. Let R := Q[β][q ±1 ij ]. Definition 4.12. Denote by 6HT n the quotient of the algebra 6T n ⊗ R by the (two-sided) ideal generated by the set of elements {r 2 i,j − β r i,j − q ij , 1 ≤ i < j ≤ n}. All these algebras are naturally graded, with deg(r i,j ) = 1, deg(β) = 1, deg(q ij ) = 2. It is clear that the algebra 6T (0) n can be considered as the infinitesimal deformation R i,j := 1 + ǫ r i,j , ǫ −→ 0, of the Yang-Baxter group 26 Y B n . Corollary 4.3. Define h ij = 1 + r ij ∈ 6HT n . Then the following relations in the algebra 6HT n are satisfied: (1) r ij r ik r jk = r jk r ik r ij for all pairwise distinct i, j and k; Note, the item (1) includes three relations in fact.

Proposition 4.3.
(1) The quadratic dual (6T n ) ! of the algebra 6T n is a quadratic algebra generated by the elements {t i,j , 1 ≤ i < j ≤ n} subject to the set of relations is a quadratic algebra with generators {t i,j , 1 ≤ i < j ≤ n} subject to the relations (ii)-(iii) above only.  The problem we are interested in, is to describe commutative subalgebras generated by additive (resp. multiplicative) Dunkl elements in the algebra 6T (0) n . Notice that the subalgebra generated by additive Dunkl elements in the abelianization 27 of the algebra 6T n (0) has been studied in [85], [78]. In order to state the result from [78] we need, let us introduce a bit of notation. As before, let Fl n denotes the complete flag variety, and denote by A n the algebra generated by the curvature of 2-forms of the standard Hermitian linear bundles over the flag variety Fl n , see e.g [78]. Finally, denote by I n the ideal in the ring of polynomials Z[t 1 , . . . , t n ] generated by the set of elements (t i 1 + · · · + t i k ) k(n−k)+1 , 26 For the reader convenience we recall the definition of the Yang-Baxter group Definition 4.13. The Yang-Baxter group Y Bn is a group generated by elements {R ±1 ij , 1 ≤ i < j ≤ n}, subject to the set of defining relations
• Hilb(6T ⋆ 4 , t) = (1, 6, 23, 65, 134, 164, 111, 43, 11, 1) t . As a consequence of the cyclic relations, one can check that for any integer n ≥ 2 the n-th power of the additive Dunkl element θ i is equal to zero in the algebra 6T ⋆ n for all i = 1, . . . , n. Therefore, the Dunkl elements generate a finite dimensional commutative subalgebra in the algebra 6T ⋆ n . There exist natural homomorphisms 3) The first and third arrows in (4.19) are epimorphism. We expect that the mapπ is also epimorphism 28 , and looking for a description of the kernel ker(π). which is a K-theoretic analog of that (4.3). It is an interesting problem to find a geometric interpretation of the algebra A mult n and the mapφ. • ("Quantization") Let β and {q ij = q ji , 1 ≤ i, j ≤ n} be parameters.  Therefore one can define algebras 6HB n and 6HA n which are a "quantum deformation" of algebras B n and A n respectively. We expect that in the case β = 0 and a special choice of "arithmetic parameters" {q ij }, the algebra HA n is connected with the Arithmetic Schubert and Grothendieck Calculi, cf [93], [85]. Moreover, for a "general"set of parameters {q ij } 1≤i,j≤n and β = 0, we expect an existence of a natural homomorphism where QK * (Fl n ) denotes a multiparameter quantum deformation of the K-theory ring K * (Fl n ), [47], [51]; see also Section 3.1. Thus, we treat the algebra HA mult n as the K-theory version of a multiparameter quantum deformation of the algebra A mult n which is generated by the curvature of 2-forms of the Hermitian linear bundles over the flag variety Fl n .

Hilbert series of algebras CY B n and 6T
In fact, the following statements are true.   (A) The Hilbert polynomial of the quadratic dual of the algebra 6T n is equal to It is well-known that where Bell n denotes the n-th Bell number, i.e. the number of ways to partition n things into subsets, see [87] Recall, that n≥0 Bell n z n n! = exp(exp(z) − 1)).
Note that Hilb(MY B n , t) = P (MY B n , −1, t) −1 and P (MY B n , 1, 1) = Bell n , the n-th Bell number.   (a + (n − k) q) t k z n n! .
(2) The even generic Orlik-Solomon algebra Definition 4.16. The even generic Orlik-Solomon algebra OS + (Γ n ) is defined to be an associative algebra (say over Z) generated by the set of mutually commuting elements y i,j , 1 ≤ i = j ≤ n, subject to the set of cyclic relations and all sequences of pairwise distinct integers 1 ≤ i 1 , . . . , i k ≤ n.
• Show that the number of degree k, k ≥ 3, relations in the definition of the Orlik-Solomon algebra OS '+ (Γ n ) is equal to 1 2 (k − 1)! n k and also is equal to the maximal number of k-cycles in the complete graph K n .
Note that if one replaces the commutativity condition in the above Definition on the condition that y i,j ′ s pairwise anticommute, then the resulting algebra appears to be isomorphic to the Orlik-Solomon algebra OS(Γ n ) corresponding to the generic hyperplane arrangement Γ n , see [79]. It is known, ibid, Corollary 5.3, that where the sum runs over all forests F on the vertices 1, . . . , n, and |F | denotes the number of edges in a forest F.

Super analogue of 6-term relations algebra
Let n, m be non-negative integers.
Remark 4.4. ("Odd" six-term relations algebra) In particular, one can define an "odd" analog 6T (−) n = 6T 0,n of the six term relations algebra 6T n . Namely, the algebra 6T (−) n is given by the set of generators {y ij , 1 ≤ i < j ≤ n}, and that of relations: 1) y i,j and y k,l anticommute if i, j, k, l are pairwise distinct; 2) [y i,j , y i,k + y j,k ] + + [y i,k , y j,k ] + = 0, if 1 ≤ i < j ≤ k ≤ n, where [x, y] + = xy + yx denotes the anticommutator of x and y.
The "odd" three term relations algebra 3T − n can be obtained as the quotient of the algebra 6T − n by the two-sided ideal generated by the three term relations y ij y jk + y jk y ki + y ki y ij = 0, if i, j, k are pairwise distinct. One can show that the Dunkl elements θ i and θ j , i = j, given by formula form an anticommutative family of elements in the algebra 6T (−) n . In a similar fashion one can define an "odd" analogue of the dynamical six term relations algebra 6DT n , see Definition 2.2 and Section 2.2, as well as define an "odd' analogues of the algebra 3HQ n (β, 0), see Definition 2.6, the Kohno-Drinfeld algebra, the Hecke algebra and few others considered in the present paper. Details are omitted in the present paper.
More generally, one can ask what are natural q-analogues of the six term and three term relations algebras ? In other words to describe relations which ensure the q-commutativity of Dunkl elements defined above. First of all it would appear natural that the "q-locality and q-symmetry conditions" hold among the set of generators {y ij , 1 ≤ i = j ≤ n}, that is y ij + q y ji = 0, y ij y kl = q y kl y ij if i < j, k < l, and {i, j} ∩ {k, l} = ∅. Another natural condition is the fulfillment of q-analogue of the classical Yang-Baxter relations, namely [y ik , y jk ] q + [y ik , y ji ] q + [Y ij , y jk ] q = 0, if i < j < k, where [x, y] q := x y − q y x denotes the q-commutator. However we are not able to find the q-analogue of the classical Yang-Baxter relation listed above in the Mathematical and Physical literature yet. Only cases q = 1 and q = −1 have been extensively studied. ij , 1 ≤ i, j ≤ n, α = 1, . . . , r}, subject to the following relations

Compatible Dunkl elements and Manin matrices
• either the unitarity (the case of sign "+"), or the symmetry relations (the case of sign " -") 30 We define global 3-term relations algebra 3T (±) n,r as " compatible product" of the local 3-term relations algebras. Namely, we require that the elements It is easy to check that our request is equivalent to a validity of the following sets of relations among the generators {u i + · · · + λ r θ also either mutually commute (the case " + ") or pairwise anticommute (the case " -"). Now we are looking for a set of relations among the local Dunkl elements which is a consequence of the commutativity (anticommutativity) of the global Dunkl elements. It is quite j ] ± , and the commutativity (or anticommutativity) of the global Dunkl elements for all (λ 1 , . . . , λ r ) ∈ R r is equivalent to the following set of relations should be either a Manin matrix (the case " + "), or its super analogue (the case " -"). Clearly enough that a similar construction can be applied to the algebras studied in Section 2, I-III.,and thus it produces some interesting examples of the Manin matrices.
It is an interesting problem to describe the algebra generated by the local Dunkl elements {θ and a commutative subalgebra generated by the global Dunkl elements inside the former. It is also an interesting question whether or not the coefficients C 1 , . . . , C n of the column characteristic polynomial Det col | Θ n − t I n |= n k=0 C k t n−k of the Manin matrix Θ n generate a commutative subalgebra ? For a definition of the column determinant of a matrix, see e.g. [16].
However a close look at this problem and the question posed needs an additional treatment and has been omitted from the content of the present paper.
Here we are looking for a "natural conditions" to be imposed on the set of generators {u α ij } 1≤α≤r 1≤i,j≤n in order to ensure that the local Dunkl elements satisfy the commutativity (or anticommutativity) relations: j ] ± = 0, f or all 1 ≤ i < j ≤ n and 1 ≤ α, β ≤ r. The "natural conditions" we have in mind are: • (twisted classical Yang-Baxter relations) if i, j, k, l are distinct and 1 ≤ α, β ≤ r.
Finally we define a multiple analogue of the three term relations algebra, denoted by 3T ± (rK n ), to be the quotient of the global 3-term relations algebra 3T ± n,r modulo the two-sided ideal generated by the left hand sides of relations (4.7), (4.8) and that of the following relations n }. The algebra (3T − (2K n )) (−) ) anti has been studied in [47], and [7]. In the present paper we state only our old conjecture. Conjecture 4.9.
(A.N. Kirillov, 2000) Hilb where for any algebra A we denote by A anti the quotient of algebra A by the two-sided ideal generated by the set of anticommutators {ab + ba | (a, b) ∈ A × A}.
According to observation of M. Haiman [37], the number 2 n (n + 1) n−2 is thought of as being equal to to the dimension of the space of triple coinvariants of the symmetric group S n .

Four term relations algebras / Kohno-Drinfeld algebras
4.3.1 Kohno-Drinfeld algebra 4T n and that CY B n Definition 4.18. The 4 term relations algebra (or the Kohno-Drinfeld algebra, or infinitesimal pure braids algebra) 4T n is an associative algebra (say over Q) with the set of generators y i,j , 1 ≤ i < j ≤ n, subject to the following relations 1) y i,j and y k,l are commute, if i, j, k, l are all distinct; Note that the algebra 4T n is given by n 2 generators and 2 n 3 + 3 n 4 quadratic relations, and the element c := 1≤i<j≤n y i,j belongs to the center of the Kohno-Drinfeld algebra.

Definition 4.19.
Denote by 4T 0 n the quotient of the algebra 4T n by the (two-sided) ideal generated by by the set of elements {y 2 i,j , 1 ≤ i < j ≤ n}. More generally, let β, {q ij , 1 ≤ i < j ≤ n} be the set of parameters, denote by 4HT n the quotient of the algebra 4T n by the two-sided ideal generated by the set of elements {y 2 ij − βy ij − q ij , 1 ≤ i < j ≤ n}.
These algebras are naturally graded, with deg(y i,j ) = 1, deg(β) = 1, deg(q ij ) = 2, as well as each of that algebras has a natural filtration by setting deg( It is clear that the algebra 4T n can be considered as the infinitesimal deformation g i,j := 1 + ǫ y i,j , ǫ −→ 0, of the pure braid group P n . There is a natural action of the symmetric group S n on the algebra 4T n ( and also on 4T 0 n ) which preserves the grading: it is defined by w · y i,j = y w(i),w(j) for w ∈ S n . The semi-direct product QS n ⋉ 4T n (and also that QS n ⋉ 4T 0 n ) is a Hopf algebra denoted by B n (respectively B 0 n ). Remark 4.5. There exists the natural map CY B n −→ 4T n , given by y i,j := u i,j + u j,i .

Indeed, one can easily check that
[y ij , y ik + y jk ] = w ijk + w jik − w kij − w kji , see Section 2.3.1, Definition 2.5 for a definition of the classical Yang-Baxter algebra CY B n , and Section 2, (2.3), for a definition of the element w ijk . Remark 4.6. It follows from the classical 3-term identity ("Jacobi identity") that if elements {y i,j | 1 ≤ i < j ≤ n} satisfy the 4-term algebra relations, see Definition 4.18, and t 1 , · · · , t n , a set of (pairwise) commuting parameters, then the elements r i,j := y i,j t i − t j satisfy the 6-term relations algebra 6T n , see Section 4.2.1" Definition 4.9. In particular, the Knizhnik-Zamolodchikov elements form a pairwise commuting family (by definition, we put y i,j = y j,i , if i > j). (2) Let A = U (sl (2)) be the universal enveloping algebra of the Lie algebra sl (2). Recall that the algebra sl (2) Then the map y i,j −→ Ω i,j ∈ A ⊗n defines a representation of the Kohno-Drinfeld algebra 4T n on that A ⊗n . The element KZ j defined above, corresponds to the truncated (or at critical level ) rational Knizhnik-Zamolodchikov operator.
where n k stands for the Stirling numbers of the second kind, i.e. the number of ways to partition a set of n things into k nonempty subsets.
Remark 4.7. It follows from [2] that Hilb(4T n , t) is equal to the generating function of Vassiliev invariants of order d for n-strand braids. Therefore, one has the following equality: i.e. the number of Vassiliev invariants of order d for n-strand braids is equal to the Stirling number of the second kind n+d−1 n−1 . We expect that the generating function of Vassiliev invariants of order d for n-strand virtual braids is equal to the Hilbert series Hilb(4N T n , t) of the nonsymmetric Kohno-Drinfeld algebra 4N T n , see Section 4.3.2.
Proposition 4.6. (Cf [3]) The algebra 4N T n , t) is Koszul, and where N (k, n) := 1 n n k n k+1 denotes the Narayana number, i.e. the number of Dyck n-paths with exactly k peaks; denotes the generalized Laguerre polynomial.
See also Theorem 4.6 below.
It is well-known that the quadratic dual 4T ! n of the Kohno-Drinfeld algebra 4T n is isomorphic to the Orlik-Solomon algebra of type A n−1 , as well as the algebra 3T anti n . However the algebra 4T 0 n is failed to be Koszul.   Definition 4.20. The nonsymmetric 4 term relations algebra (or the nonsymmetric Kohno-Drinfeld algebra) 4N T n is an associative algebra (say over Q) with the set of generators y i,j , 1 ≤ i = j ≤ n, subject to the following relations 1) y i,j and y k,l are commute, if i, j, k, l are all distinct; 2) [y i,j , y i,k + y j,k ] = 0, if i, j, k are all distinct.
We denote by 4N T + n the quotient of the algebra 4N T n by the two-sided ideal generated by the elements {y ij + y ji = 0, 1 ≤ i = j ≤ n}. for all n ≥ 2.
We expect that the both algebras 4N T n and 4N T + n are Koszul.
(1) Define the McCool algebra PΣ n to be the quotient of the nonsymmetric Kohno-Drinfeld algebra 4N T n by the two-sided ideal generated by the elements {y ik y jk − y jk y ik } for all pairwise distinct i, j and k.
(2) Define the upper triangular McCool algebra PΣ + n to be the quotient of the McCool algebra PΣ n by the two-sided ideal generated by the elements (1 + jt).

Proposition 4.7.
(1) The quadratic dual PΣ ! n of the algebra PΣ n admits the following description. It is generated over Z by the set of pairwise anticommuting elements subject to the set of relations (a) y 2 ij = 0, y ij y ji = 0, 1 ≤ i = j ≤ n, (b) y ik y jk = 0, for all distinct i, j, k, (c) y ij y jk + y ik y ij + y kj y ik = 0, for all distinct i, j, k.
(2) The quadratic dual (PΣ + n ) ! of the algebra PΣ + n admits the following description. It is generated over Z by the set of pairwise anticommuting elements {z ij , 1 ≤ i < j ≤ n}, subject to the set of relations (a) z 2 ij = 0 for all i < j, (b) z ij z jk = z ij z ik for all 1 ≤ i < j < k ≤ n.

Comments 4.5. The McCool groups and algebras
The McCool group P Σ n is by definition, the group of pure symmetric automorphisms of the free group F n consisting of all automorphism that, for a fixed basis {x 1 , . . . , x n }, send each x i to a conjugate of itself. This group is generated by automorphisms α ij , 1 ≤ i = j ≤ n, defined by

McCool have proved that the relations
form the set of defining relations for the group P Σ n The subgroup of P Σ n generated by the α ij for 1 ≤ i < j ≤ n is denoted by P Σ + n and is called by upper triangular McCool group. It is easy to see that the McCool algebras PΣ n and PΣ + n are the " infinitesimal deformations " of the McCool groups P Σ n and P Σ + n respectively. (I) Algebra 4T T n is generated over Z by the set of elements {x ij , 1 ≤ i = j ≤ n}, subject to the set of relations (II) Algebra 4ST n is generated over Z by the set of elements {x ij , 1 ≤ i = j ≤ n}, subject to the set of relations Therefore, dim(4T T n ) ! is equal to the number of permutations of the set [1, ..., n + 1] having no substring [k, k+1]; also, for n ≥ 1 equals to the maximal permanent of a nonsingular n×n (0, 1)matrix, see [87], A000255 31  We expect that The both algebras 4T T n and 4ST n are Koszul.
Problem. Give a combinatorial interpretation of polynomials Hilb((4T T n ) ! , t) and construct a monomial basis in the algebras (4T T n ) ! and 4ST n .  1 0 The Jucys-Murphy elements d j , 2 ≤ j ≤ n, commute pairwise in the algebra 4T n . 2 0 In the algebra 4T 0 n the Jucys-Murphy elements d j , 2 ≤ j ≤ n, satisfy the following relations

Subalgebra generated by Jucys-Murphy elements in
3 0 Subalgebra (over Z) in 4T 0 n generated by the Jucys-Murphy elements d 2 , · · · , d n has the following Hilbert polynomial  (1) It is clearly seen that the commutativity of the Jucys-Murphy elements is equivalent to the validity of the Kohno-Drinfeld relations and the locality relations among the generators {y i,j } 1≤i<j≤n .
(2) Let's stress that d 2j−2 j = 0 in the algebra 4T 0 n , for j = 3, . . . , n. For example, d 4 3 = y 13 y 23 y 13 y 23 + y 23 y 13 y 23 y 13 = 0 since dim(4T 0 3 ) 4 = 1 and it is generated by the element d 4 3 . (3) The map ι : y i,j −→ y n+1−j,n+1−i preserves the relations 1) and 2) in the definition of the algebra 4T n , and therefore defines an involution of the Kohno-Drinfeld algebra. Hence the elements also form a pairwise commuting family. It is well-known that the Kohno-Drinfeld algebra 4T n is Koszul, and its quadratic dual 4T ! n is isomorphic to the anticommutative quotient 3T 0,anti n of the algebra 3T On the other hand, if n ≥ 3 the algebra 4T 0 n is not Koszul, and its quadratic dual is isomorphic to the quotient of the ring of polynomials in the set of anticommutative variables {t i,j | 1 ≤ i < j ≤ n}, where we do not impose conditions t 2 ij = 0, modulo the ideal generated by Arnold's relations {t i,j t j,k + t i,k (t i,j − t j,k ) = 0} for all pairwise distinct i, j and k.

Nonlocal Kohno-Drinfeld algebra NL4T n
Definition 4.24. Nonlocal Kohno-Drinfeld algebra N L4T n is an associative algebra over Z with the set of generators {y ij , 1 ≤ i < j ≤ n} subject to the set of relations  (1) The algebra N L4T n is Koszul, and where C k = 1 k+1 2k k stands for the k-th Catalan number. (2) The quadratic dual (N L4T n ) ! of the nonlocal Kohno -Drinfeld algebra N L4T n is an associative algebra generated by the set of mutually anticommuting elements {t ij 1 ≤ i < j ≤ n} subject to the set of relations Therefore the algebra (N L4T n ) ! is the quotient of the the Orlik-Solomon algebra OS n by the ideal generated by Disentanglement relations, and dim((N L4T n+1 ) ! ) is equal to the number of Schroeder paths , i.e. paths from (0, 0) to (2n, 0) consisting of steps U = (1, 1), D = (1, −1), H = (2, 0) and never going below the x − axis. The Hilbert polynomial Hilb((N L4T n ) ! , t) is the generating function of such paths with respect to the number of U ′ s, see [87], A088617.

Remark 4.10.
Denote by H n (q) "the normalized" Hecke algebra of type A n , i.e. an associative algebra generated over Z[q, q −1 ] by elements T 1 , . . . , T n−1 subject to the set of relations (a) Therefore the map y ij → H (ij) defines a epimorphism ι n : N L4T n −→ H n+1 (q).
Definition 4.25. Denote by N L4T n the quotient of the non-local Kohno-Drinfeld algebra N L4T n by the two-sided ideal I n generated by the following set of degree three elements: (1) z ij := y i,j+1 y ij y j,j+1 − y j,j+1 y ij y i,j+1 , if 1 ≤ i < j ≤ n, Proposition 4.9.
Therefore, there exists an epimorphism of algebras N L4T n −→ H n (q), and images of the elements e k (d 2 , . . . , d n ), (resp. e k (d 2 , . . . ,d n ) 1 ≤ k < n, belongs to the center of the "normalized" Hecke algebra H n (q), and in fact generate the center of algebra H n (q).
Few comments in order: (A) Let N ℓ4T n be an associative algebra over Z with the set of generators {y ij , 1 ≤ i < j ≤ n} subject to the set of relations [y ij , j a=i y ak ] = 0, if i < j < k. Proposition 4.10.
(1) The algebra N ℓ4T n is Koszul and has the Hilbert series equals to Hilb(N ℓ4T n , t) = ( where N (k, n) := 1 n n k n k+1 denotes the Narayana number, i.e. the number of Dyck n-paths with exactly k peaks, see e.g. [87], A001263. Therefore, dim(N ℓ4T n ) ! = 1 , as well as the following set of commutators It is an interesting task to find defining relations among the Jucys-Murphy elements {d j , j = 2, . . . , n} in the algebra N L4T n or that N ℓ4T n . We expect that the Jucys-Murphy element d k satisfies the following relation (= minimal polynomial) in the Hecke algebra H n (q), n ≥ k,

On relations among JM-elements in Hecke algebras
Let H n (q) be the "normalized" Hecke algebra of type A n , see Remark 4.10. Let λ ⊢ n be a partition of n. For a box x = (i, j) ∈ λ define It is clear that if q = 1, c q=1 (x) is equal to the content c(x) of a box x ∈ λ. Denote by  such that for any partition λ ⊢ n one has f (c λ (x; q) | x ∈ λ) = 0.
For example, one can check that symmetric polynomial belongs to the set J q . Finally, denote by J  It seems an interesting problem to find a minimal set of generators for the ideal J Comments 4.6. Denote by JM (n) the algebra over Z generated by the JM-elements d 2 , . . . , d n , deg(d i = 1, ∀i, corresponding to the symmetric group S n . In this case one can check Conjecture 8 for n < 8, and compute the Hilbert polynomial(s) of the associated graded algebra(s) gr(JM (n)). It seems an interesting task to find a combinatorial interpretation of the polynomials Hilb(gr(JM (n)), t) in terms of standard Young tableaux of size n.
Let {χ λ , λ ⊢ n} be the characters of the irreducible representations of the symmetric group S n , which form a basis of the center Z n of the group ring Z[S n ]. The famous result by A. Jucys [40] states that for any symmetric polynomial f (z 1 , . . . , z n ) the character expansion of f (d 2 , . . . , d n , 0) ∈ Z n is where H λ = x∈λ h x denotes the product of all hook-lengths of λ, and C λ := {c(x)} x∈λ denotes the set of contents of all boxes of λ.
Recall that the Jucys-Murphy elements {d H j } 2≤j≤n in the (normalized) Hecke algebra H n (q) are defined as follows: d H j := i<j T (ij) , where T (ij) := T i · · · T j−1 T j T j−1 · · · T i . Finally denote by H λ (q) and C (q) λ the hook polynomial and the set {c λ x; q)} x ∈ λ. Then for any symmetric polynomial f (z 1 , . . . , z n ) one has where chi λ q denotes the q-character of the algebra H n(q) . Therefore, if f ∈ J 4.6 Extended nil-three term relations algebra and DAHA, cf [15] Let A := {q, t, a, b, c, h, e, f, . . .} be a set of parameters.
Note that the algebra 3T n contains also the set of elements {π a u jn , 1 ≤ a ≤ n − j}.
Each of the set of elements by itself generate the symmetric group S n .
Comments 4.7. Let A = (a, b, c, h, e) be a sequence of integers satisfying the conditions (4.5).
Denote by ∂ A i the divided difference operator It follows from Lemma 4.5 that the operators {∂ A i } 1≤i≤n satisfy the Coxeter relations Definition 4.29.
(1) Let w ∈ S n be a permutation. Define the generalized Schubert polynomial corresponding to permutation w as follows and w 0 denotes the longest element in the symmetric group S n .
(2) Let α be a composition with at most n parts, denote by w α ∈ S n the permutation such that w α (α) = α, where α denotes a unique partition corresponding to composition α. Proposition 4.11. ( [46]) Let w ∈ S n be a permutation.
In all cases listed above the polynomials S A w (X n ) have non-negative integer coefficients. • If A = (1, −1, 1, −h, 0), then S A w (X n ) is equal to the h-Schubert polynomials introduced in [46]. Define the generalized key or Demazure polynomial corresponding to a composition α as follows • If A = (1, 0, 1, 0, 0), then K A α (X n ) is equal to key (or Demazure) polynomial corresponding to α.
In all cases listed above the polynomials S A w (X n ) have non-negative integer coefficients.
(1) Let b, c, h, e be a collection of integers, define elements Show that (2) Assume that a = q, b = −q, c = q −1 , h = e = 0, and introduce elements (a) Show that if i, j, k are distinct, then (b) Assume additionally that u ij u jk u ij = 0, if i, j, k are distinct.

Remark 4.11.
Let us stress on a difference between elements T ij as a part of generators of the algebra 3T n , and the elements T (ij) := T i · · · T j−1 T j T j−1 · · · T i ∈ H n (q).
Whereas one has [T ij , T kl ] = 0, if i, j, k, l are distinct, the relation [T (ij) , T (kl) ] = 0 in the algebra H n (q) holds (for general q and i ≤ k) if and only if either one has i < j < k < l, or i < k < l < j. (1) Definition 4.30.
It is well-known that the group SL(2, Z) can be generated by two matrices which satisfy the following relations Let us introduce operators τ + and τ − acting on the extended affine algebra H n . Namely, In the last formula we set T n = 1 for convenience.

Combinatorics of associative Yang-Baxter algebras
Let α and β be parameters. (1) The associative quasi-classical Yang-Baxter algebra of weight (α, β), denoted by ACY B n (α, β), is an associative algebra, over the ring of polynomials Z[α, β], generated by the set of elements {x ij , 1 ≤ i < j ≤ n}, subject to the set of relations (2) Define associative quasi-classical Yang-Baxter algebra of weight β ,denoted by ACY B n (β), to be ACY B n (0, β).
The algebra 3T n (β), see Definition 3.1, is the quotient of the algebra ACY B n (−β), by the "dual relations" x jk x ij − x ij x ik − x ik x jk + β x ik = 0, i < j < k.
The (truncated) Dunkl elements θ i = j =i x ij , i = 1, . . . , n, do not commute in the algebra ACY B n (β). However a certain version of noncommutative elementary polynomial of degree k ≥ 1, still is equal to zero after the substitution of Dunkl elements instead of variables, [47]. We state here the corresponding result only "in classical case", i.e. if β = 0 and q ij = 0 f or all i, j.

Combinatorics of Coxeter element
Consider the "Coxeter element" w ∈ ACY B n (α, β) which is equal to the ordered product of "simple generators": x a,a+1 .
Let us bring the element w to the reduced form in the algebra ACY B n (α, β), that is, let us consecutively apply the defining relations (a) and (b) to the element w in any order until unable to do so. Denote the resulting (noncommutative) polynomial by P n (x ij ; α, β). In principal, the polynomial itself can depend on the order in which the relations (a) and (b) are applied. We set P n (x ij ; β) := P n (x ij ; 0, β). (1) Apart from applying the relation (a) (commutativity), the polynomial P n (x ij ; β) does not depend on the order in which relations (a) and (b) have been applied, and can be written in a unique way as a linear combination: where the second summation runs over all sequences of integers {i a } s a=1 such that n − 1 ≥ i 1 ≥ i 2 ≥ . . . ≥ i s = 1, and i a ≤ n − a for a = 1, . . . , s − 1; moreover, the corresponding sequence {j a } n−1 a=1 can be defined uniquely by that {i a } n−1 a=1 . • It is clear that the polynomial P (x ij ; β) also can be written in a unique way as a linear combination of monomials s a=1 x ia,ja such that j 1 ≥ j 2 . . . ≥ j s .
(4) Upon the specialization x 1j −→ t, 1 ≤ j ≤ n, and that x ij −→ 1, if 2 ≤ i < j ≤ n, the polynomial P (x ij ; β) is transformed to the polynomial where the second summation runs over the set of Dick paths π of length 2n with exactly k picks (UD-steps), and p(π) denotes the number of valleys (DU-steps) that touch upon the line x = 0.
In particular, where N (n, k) denotes the Narayana numbers, see item (3) of Proposition 5.1. More generally, write P n (t, β) = k P (k) Comments 5.2.
• Note that if β = 0, then one has G (β=0) w (x 1 , . . . , x n−1 ) = S w (x 1 , . . . , x n−1 ), that is the β-Grothendieck polynomial at β = 0, is equal to the Schubert polynomial corresponding to the same permutation w. Therefore, if π= 1 2 3 . . . n 1 n n − 1 . . . 2 , then S π (x 1 = 1, . . . , t n−1 = 1) = C n−1 , where C m denotes the m-th Catalan number. Using the formula (5.20) it is not difficult to check that the following formula for the principal specialization of the Schubert polynomial S π (X n ) is true where C m (q) denotes the Carlitz -Riordan q-analogue of the Catalan numbers, see e.g. [88]. The formula (5.20) has been proved in [29] using the observation that π is a vexillary permutation, see [61] for the a definition of the latter. A combinatorial/bijective proof of the formula (5.20) is is due to A.Woo [98].
Show that Note that the number n − k + 1 n + 1 n + k k is equal to the dimension of irreducible representation of the symmetric group S n+k that corresponds to partition (n + k, k).

Multiparameter deformation of Catalan, Narayana and Schröder numbers
Let b = (β 1 , . . . , β n−1 ) be a set of mutually commuting parameters. We define a multiparameter analogue of the associative quasi-classical Yang-Baxter algebra M ACY B n (b) as follows.

Definition 5.2. (Cf Definition 2.4)
The multiparameter associative quasi-classical Yang-Baxter algebra of weight b, denoted by M ACY B n (b), is an associative algebra, over the ring of polynomials Z[β 1 , . . . , β n−1 ], generated by the set of elements {x ij , 1 ≤ i < j ≤ n}, subject to the set of relations Consider the "Coxeter element" w n ∈ M ACY B n (b) which is equal to the ordered product of "simple generators": x a,a+1 . Now we can use the same method as in [90], 8.C5, (c) , see Section 5.1, to define the reduced form of the Coxeter element w n . Namely, let us bring the element w n to the reduced form in the algebra M ACY B n (b), that is, let us consecutively apply the defining relations (a) and (b) to the element w n in any order until unable to do so. Denote the resulting (noncommutative) polynomial by P (x ij ; b). In principal, the polynomial itself can depend on the order in which the relations (a) and (b) are applied. To state our main result of this Subsection, let us define polynomials Q(β 1 , . . . , β n−1 ) := P (x ij = 1, ∀i, j ; β 1 − 1, β 2 − 1, . . . , β n−1 − 1).
It follows from [90] and Proposition 5.1, that Polynomials Q(β 1 , . . . , β n−1 ) and Q(β 1 + 1, . . . , β n−1 + 1) can be considered as a multiparameter deformation of the Catalan and (small) Schröder numbers correspondingly, and the homogeneous degree k part of Q(β 1 , . . . , β n−1 ) as a multiparameter analogue of Narayana numbers. If p is a Schröder path, we denote by d(p) the number of the diagonal steps resting on the path p, and by a(p) the number of unit squares located between the path p and the diagonal x = y. For each (unit) diagonal step D of a path p we denote by i(D) the x-coordinate of the column which contains the diagonal step D. Finally, define the index i(p) of a path p as the some of the numbers i(D) for all diagonal steps of the path p. where the sum runs over the set of all Schröder paths of length n.

Comments 5.3.
The q-Schröder polynomials defined by the formula (5.22) are different from the q-analogue of Schröder polynomials which has been considered in [11]. It seems that there are no simple connections between the both. Proposition 5.3.
(2) In the cases when β = 1 and 0 ≤ n − k ≤ 2, the value of the determinant in the RHS(5.8) is known, see e.g. [11], or M. Ichikawa talk Hankel determinants of Catalan, Motzkin and Schrder numbers and its q-analogue, http:/denjoy.ms.u-tokyo.ac.jp. One can check that in the all cases mentioned above, the formula (5.8) gives the same results.
(3) Grothendieck and Narayana polynomials It follows from the expression (5.7) for the Narayana-Schröder polynomials that P n (β − 1) = N n (β), where It is well-known, see e.g. [92], that the Narayana polynomial N n (β) is equal to the generating function of the statistics π(p) = (number of peaks of a Dick path p) − 1 on the set Dick n of Dick paths of the length 2n N n (β) = p β π(p) .
We denote the sum in the RHS(5.9) by N (k) is a symmetric polynomial in β with non-negative integer coefficients, and .

Grothendieck polynomials and k-dissections
Let k ∈ N and n ≥ k − 1, be a integer, define a k-dissection of a convex (n + k + 1)-gon to be a collection E of diagonals in (n + k + 1)-gon not containing (k + 1)-subset of pairwise crossing diagonals and such that at least 2(k − 1) diagonals are coming from each vertex of the (n + k + 1)gon in question. One can show that the number of diagonals in any k-dissection E of a convex (n + k + 1)-gon contains at least (n + k + 1)(k − 1) and at most n(2k the generating function for the number of k-dissections with a fixed index, where the above sum runs over the set of all k-dissections of a convex (n + k + 1)-gon.
For the permutation w (n) defined above, one has the following factorization formula for the Grothendieck polynomial corresponding to w (n) , [61], In particular, if then the principal specialization G In particular, the polynomial G (β−1) w (n) (x 1 , . . . , x n ) is a symmetric polynomial in β with nonnegative integer coefficients.
Exercises 5.4. Let 1 ≤ k ≤ m ≤ n be integers, n ≥ 2k + 1. Consider permutation Show that S w (1, q, . . .) = q n(D(w)) C (m) where for any permutation w, n(D(w)) = d i (w) 2 and d i (w) denotes the number of boxes in the i-th column of the (Rothe ) diagram D(w) of the permutation w, see [61]. p.8.

(C) A determinantal formula for the Grothendieck polynomials
Theorem 5.7.
(a) One can compute the Grothendieck polynomials for yet another interesting family of permutations. namely, grassmannian permutations where s (n,1 j ) (X k ) denotes the Schur polynomial corresponding to the hook shape partition (n, 1 j ) and the set of variables X k := (x 1 , . . . , x k ). In particular,

(b) Grothendieck polynomials for grassmannian permutations
In the case of a grassmannian permutation w := σ λ ∈ S ∞ of the shape λ = (λ 1 ≥ λ 2 ≥ . . . ≥ λ n ) where n is a unique descent of w, one can prove the following formulas for the β-Grothendieck polynomial where X [i,n] = (x i , x i+1 , . . . , x n ), and for any set of variables X, and h k (X) denotes the complete symmetric polynomial of degree k in the variables from the set X.
A proof is a straightforward adaptation of the proof of special case β = 0 (the case of Schur polynomials) given by I. Macdonald [61], Section 2, (2.10) and Section 4, (4.8).
Indeed, consider β-divided difference operators π (β) j , j = 1, . . . , n − 1, and π (β) w , w ∈ S n , introduced in [27]. For example, be the longest element in the symmetric group S n . The same proves of the statements (2.10), (2.16) from [61] show that On the other hand, the same arguments as in the proof of statement (4.8) from [61] show that . Application of the formula for operator π (β) w (0) n displayed above to the monomial x λ+δn finishes the proof of the first equality in (5.11). The statement that the right hand side of the equality (5.12) coincides with determinants displayed in the identity (5.12) can be checked by means of simple transformations..
(2) Let w ∈ S n be a permutation, and CS(w) be the set of compatible sequences corresponding to w, see e.g. [8].
Define statistics c(•) on the set CS(w) such that (3) Let w be a vexillary permutation. Find a determinantal formula for the β-Grothendieck polynomial G For example, let w ∈ S n be an involution, i.e. w 2 = 1, and w ′ ∈ S n+1 be the image of w under the natural embedding S n ֒→ S n+1 given by w ∈ S n −→ (w, n + 1) ∈ S n+1 . It is well-known, see e.g. [53], [98], that the multiplicity m e,w of the 0-dimensional Schubert cell {pt} = Y w (n+1) 0 in the Schubert variety Y w ′ is equal to the specialization S w (x i = 1) of the Schubert polynomial S w (X n ). Therefore one can consider the polynomial S Question What is a geometrical meaning of the coefficients of the polynomial S (β) is a unimodal polynomial for any permutation w.
We denote the above determinant by D(n, k, r, b, p).
It is convenient to re-wright the above formula for D(n, k, r, b, p) in the following form We consider below some special cases of Theorem 5.7 in the case r = 1. To simplify notation, we set D(n, k, b, p) := D(n, k, r = 1, b, p). Then we can rewrite the above formula for D(n, k, r, b, p) as follows In particular, denotes the generalized Fuss-Catalan number.
See [87], A005700 for several combinatorial interpretations of these numbers.
(2) (R.A. Proctor [82]) Consider the Young diagram For each box (i, j) ∈ λ define the numbers c(i, j) := n + 1 − i + j, and T hen Therefore, D(n, k, b, p) is a polynomial in k with rational coefficients.
Exercises 5.5. Show that Clearly that if b = 0, then F n (0) = C n , and D(n, k, 0, 1) is equal to the Catalan-Hankel determinant C (k) n . Finally we recall that the generalized Fuss-Catalan number F (p+1) n+1 (b) counts the number of lattice paths from (0, 0) to (b + np, n) that do not go above the line x = py, see e.g. [55].

Comments 5.6.
It is well-known, see e.g. [82], or [88], vol.2, Exercise 7.101.b, that the number D(n, k, b, p) is equal to the total number pp λ n,p,b (k) of plane partitions 33 bounded by k and contained in the shape λ n,b,p .
More generally, see e.g. [29], for any partition λ denote by w λ ∈ S ∞ a unique dominant permutation of shape λ, that is a unique permutation with the code c(w) = λ. Now for any 33 Let λ be a partition. A plane (ordinary) partition bounded by d and shape λ is a filling of the shape λ by the numbers from the set {0, 1, . . . , d} in such a way that the numbers along columns and rows are weakly decreasing.
A reverse plane partition bounded by d and shape λ is a filling of the shape λ by the numbers from the set {0, 1, . . . , d} in such a way that the numbers along columns and rows are weakly increasing.
non-negative integer k consider the so-called shifted dominant permutation w (k) λ which has the shape λ and the flag φ = (φ i = k + i − 1, i = 1, . . . , ℓ(λ)). Then where pp λ (≤ k) denotes the number of all plane partitions bounded by k and contained in λ. Moreover, where P P λ (≤ k) denotes the set of all plane partitions bounded by k and contained in λ.
(2) Let λ = ((n + ℓ) ℓ , ℓ n ) be a fat hook. Show that where a(ℓ, n) is a certain integer we don't need to specify in what follows; denotes the MacMahon generating function for the number of plane partitions fit inside the box N × N × ℓ ; K λ (q) is a polynomial in q such that K λ (0) = 1.
(a) Show that where M (a, b, c) denotes the number of plane partitions fit inside the box a×b×c. It is well-known, see e.g. [62], p. 81, that • K λ (q) = π∈B n,n,ℓ where the sum runs over the set of plane partitions π = (π ij ) 1≤i,j≤n fit inside the box B n,n,ℓ := n × n × ℓ, and (c) Assume as before that λ := ((n + ℓ) ℓ , ℓ n ). Show that where the sum runs over the set of partitions µ with the number of parts at most ℓ, and Therefore the generating function P P (ℓ,0) (q) := π∈P P (ℓ,0) q |π| is equal to (5.14) where µ ′ denotes the conjugate partition of µ, therefore n(µ ′ ) = i≥1 µ i 2 . The formula (5.14) is the special case n = m of Theorem 1.2, [72].
Finally we observe that if k = np + 1, then where the numbers A (p) n are integers that generalize the numbers of alternating sign matrices (ASM) of size n × n, recovered in the case p = 2, see [74], [19] for details.
One can compute the principal specialization of the Schubert polynomial corresponding to the transposition t k,n := (k, n − k) ∈ S n that interchanges k and n − k, and fixes all other elements of [1, n].
Show that the polynomial S w (x i = 1, ∀i) has degree n(n − 1) and the coefficient is equal to the n-th Catalan number C n .
More generally, consider permutation w (n) k := 1 k × w (n) ∈ S k+2n+1 , and polynomials The polynomials P k (z) are well-known as Swiss-Knife polynomials, see [87], A153641, where one can find an overview of some properties of the Swiss-Knife polynomials.
• Show that the Grothendieck polynomial • Find a combinatorial interpretations of polynomial G λ (β).
Final remark, it follows from the seventh exercise listed above, that the polynomials S q, x j = 1, ∀j ≥ 2) define a (q, β)-deformation of the number V SASM (k) (the case σ + k ) and the number CST CP P (k) (the case σ − k ), respectively.

Specialization of Grothendieck polynomials
Let p, b, n and i, 2i < n be positive integers. Denote by T where N E(p) is equal to the number of steps N E resting on path p; E in (p) is equal to 1, if the path p starts with step E and 0 otherwise; N end (p) is equal to 1, if the path p ends by the step N and 0 otherwise.

5.3
The "longest element" and Chan-Robbins-Yuen polytope in any order until unable to do so. Denote the resulting polynomial by Q n (x ij ; α, β). Note that the polynomial itself depends on the order in which the relations (a ′ ) and (b) are applied. We denote by Q n (β) the specialization x ij = 1 for all i and j, of the polynomial Q n (x ij ; α = 0, β).
Here for any integral convex polytope P ⊂ Z d , ι(P, n) denotes the number of integer points in the set nP ∩ Z d .
In particular, the polynomial Q n (β) does not depend on the order in which the relations (a ′ ) and (b) have been applied. Now let us denote by Q n (t; α, β) the specialization x ij = 1, i < j < n, and x i,n = t, if i = 1, . . . , n − 1, of the (reduced) polynomial Q n (x ij ; α, β) obtained by applying the relations (a ′ ) and (b) in a certain order. The polynomial itself depends on the order selected. ]. Then c (dn) n (1) = a 2 n f or some non − negative integer a n . Moreover, there exists a polynomial a n (t) ∈ N[t] such that c (dn) n (t) = a n (1) a n (t), a n (0) = a n−1 .
(C) The all roots of the polynomial Q n (β) belong to the set R <−1 .
Question Let N be a positive integer. Does there exist a vexillary (grassmannian ?) permutation w ∈ S n such that n ≤ 2κ(N ) and S w (1) = N ?

Then
(1) Let R a (t 1 , . . . , t n−1 , α, β) be the following specialization x ij −→ t j−1 f or all 1 ≤ i < j ≤ n of the reduced polynomial R a (x ij ) of monomial M a ∈ ACY B n (α, β). Then the polynomial R a (t 1 , . . . , t n−1 , α, β) is well-defined, i.e. does not depend on an order in which relations (a ′ ) and (b) , Definition 5.1, have been applied.
(3) Show that polynomial R n (t, 1) has degree e n := (n + 1)(n − 2)/2, and (1) Assume additionally to the conditions (a ′ ) and (b) above that What one can say about a reduced form of the element w 0 in this case ?
Does there exist "a natural" bijection between the primitive factorizations and monomials which appear in the polynomial Q n (x ij ; β) ?
(1) Show that if n ≥ m, then (2) Show that if n ≥ m ≥ k, then P In particular, if n ≥ m ≥ k, then Note that the set of relations from the item (1) allows to give an explicit formula for the polynomial P M (β) for any dominant sequence M = (m 1 ≥ m 2 ≥ . . . ≥ m k ) ∈ (Z >0 ) k . Namely, P M (β + 1) = a k j=2 m j + a j−1 − 1 where the first sum runs over the following {0 ≤ b j ≤ min(m j+1 , m j − a j + a j−1 )}, j = 1, . . . , k − 1.
(5) Let T ∈ ST Y ((n + k, k)) be a standard Young tableau of shape (n + k, k). Denote by r(T ) the number of integers j ∈ [1, n + k] such that the integer j belongs to the second row of tableau T, whereas the number j + 1 belongs to the first row of T. Show that P x n 12 x 23 ···x k+1,k+2 (β − 1) = T ∈ST Y ((n+k,k)) β r(T ) .  Show that polynomials R n (t) have non-negative coefficients, and R n (0) = (3n) ! 6 (n !) 3 .

Conjecture 5.12.
Let λ be a partition. The element s λ (θ  (1) Define a bijection between monomials of the form s a=1 x ia,ja involved in the polynomial P (x ij ; β), and dissections of a convex (n + 2)-gon by s diagonals, such that no two diagonals intersect their interior.
(2) Describe permutations w ∈ S n such that the Grothendieck polynomial G w (t 1 , . . . , t n ) is equal to the "reduced polynomial" for a some monomial in the associative quasi-classical Yang-Baxter algebra ACY B n (β). ?
One can compare these formulas for polynomials P s ab (x ij = 1; β) with those for the β-Grothendieck polynomials corresponding to transpositions (a, b), see Comments 5.5.

Appendix I Grothendieck polynomials
Definition 6.1.
Let β be a parameter. The Id-Coxeter algebra IdC n (β) is an associative algebra over the ring of polynomials Z[β] generated by elements e 1 , . . . , e n−1 subject to the set of relations • e i e j = e j e i , if i − j ≥ 2, • e i e j e i = e j e i e j , if i − j = 1, • e 2 i = β e i , 1 ≤ i ≤ n − 1.
It is well-known that the elements {e w , w ∈ S n } form a Z[β]-linear basis of the algebra IdC n (β). Here for a permutation w ∈ S n we denoted by e w the product e i 1 e i 2 · · · e i ℓ ∈ IdC n (β), where (i 1 , i 2 , . . . , i ℓ ) is any reduced word for a permutation w, i.e. w = s i 1 s i 2 · · · s i ℓ and ℓ = ℓ(w) is the length of w.
Let x 1 , x 2 , . . . , x n−1 , x n = y, x n+1 = z, . . . be a set of mutually commuting variables. We assume that x i and e j commute for all values of i and j. Let us define h i (x) = 1 + xe i , and A i (x) = i a=n−1 h a (x), i = 1, . . . , n − 1.
One has (1) (Addition formula) where we set (x ⊕ y) := x + y + βxy; (2) (Yang-Baxter relation) The second equality follows from the first one by induction using the Addition formula, whereas the fist equality follows directly from the Yang-Baxter relation. Definition 6.2.
It is easy to see that the Jacobi matrix corresponding to the set of polynomials g r (X n 1 ,...,n k−1 ) n k ≤ r ≤ n, has nonzero determinant, and the component of maximal degree n max := l<j n i n j in the ring QH * (F n 1 ,··· ,n k , Z) is a Z[q 1 , . . . , q k−1 ]−module of rank one with generator Therefore, one can define a scalar product (the Grothendieck residue) •, • : HQ * (F n 1 ,··· ,n k , Z) × HQ * (F n 1 ,··· ,n k , Z) −→ Z[a 1 , . . . , q k−1 ] setting for elements f and g of degrees a and b, f, h = 0, if a + b = n max , and f, h = λ(q), if a + b = n max and f h = λ(q) Λ. It is well known that the Grothendieck pairing •, • is nondegenerate (for any choice of parameters q 1 , . . . , q k−1 ). Finally we state "a mirror presentation" of the small quantum cohomology ring of partial flag varieties. To start with, let n = n 1 + . . . + n k , k ∈ Z ge2 be a composition of size n, and consider the set Σ(n) = {(i, j) ∈ Z × Z |1 ≤ i ≤ N a , M a+1 + 1 ≤ j ≤ M a , a = 1, . . . , k − 1}, where N a = n 1 + . . . + n a , N 0 = 0, N k = n M a = n a+1 + . . . + n k , M 0 = n, M k = 0.
With these data given, let us introduce the set of variables