Notes on Schubert, Grothendieck and Key Polynomials

We introduce common generalization of (double) Schubert, Grothendieck, Demazure, dual and stable Grothendieck polynomials, and Di Francesco-Zinn-Justin polynomials. Our approach is based on the study of algebraic and combinatorial properties of the reduced rectangular plactic algebra and associated Cauchy kernels.


Introduction
The Grothendieck polynomials had been introduced by A. Lascoux and M.-P. Schützenberger in [29] and studied in detail in [37]. There are two equivalent versions of the Grothendieck polynomials depending on a choice of a basis in the Grothendieck ring K ⋆ (F l n ) of the complete flag variety F l n . The basis {exp(ξ 1 ), . . . , exp(ξ n )} in K * (F l n ) is a one choice, and another choice is the basis {1 − exp(−ξ j ), 1 ≤ j ≤ n}, where ξ j , 1 ≤ j ≤ n} denote the Chern classes of the tautological linear bundles L j over the flag variety F l n . In the present paper we use the basis in a deformed Grothendieck ring K * ,β (F n ) of the flag variety F l n generated by the set of elements {x i = x (β) i = 1 − exp(β ξ i ), i = 1, . . . , n}. This basis has been introduced and used for construction of the β-Grothendieck polynomials in [8], [9].
A basis in the classical Grothendieck ring of the flag variety in question corresponds to the choice β = −1. For arbitrary β the ring generated by the elements {x (β) i , 1 ≤ i ≤ n} has been identified with the Grothendieck ring corresponding to the generalized cohomology theory associated with the multiplicative formal group law F (x, y) = x + y + β x y, see [15]. The Grothendieck polynomials corresponding to the classical K-theory ring K ⋆ (F l n ), i.e. the case β = −1, had been studied in depth by A. Lascoux and M.-P. Schützenberger in [30]. The β-Grothendieck polynomials has been studied in [8], [10], [15].
The plactic monoid over a finite totally ordered set A = {a < b < c < . . . < d} is the quotient of the free monoid generated by elements from A subject to the elementary Knuth transformations [21] bca = bac & acb = cab, and bab = bba & aba = baa, (1.1) for any triple {a < b < c} ⊂ A. To our knowledge, the concept of "plactic monoid" has its origins in a paper by C.Schensted [52], concerning the study of the longest increasing subsequence of a permutation, and a paper by D. Knuth [21], concerning the study of combinatorial and algebraic properties of the Robinson-Schensted correspondence 1 .
As far as we know, this monoid and the (unital) algebra P(A) corresponding to that monoid 2 , had been introduced, studied and used in [53], Section 5, to give the first complete proof of the famous Littlewood-Richardson rule in the theory of Symmetric functions. A bit later this monoid, was named the "monoïde plaxique" and studied in depth by A. Lascoux and M.-P. Schützenberger [28]. The algebra corresponding to plactic monoid is commonly known as plactic algebra. One of the basic properties of the plactic algebra [53] is that it contains the distinguish commutative subalgebra which is generated by noncommutative elementary symmetric polynomials e k (A n ) = i 1 >i 2 >...>i k a i 1 a i 2 · · · a i k , k = 1, . . . , n, (1.2) see e.g. [53], Corollary 5.9, [7]. We refer the reader to nice written overview [40] of the basic properties and applications of the plactic monoid in Combinatorics.
It is easy to see that the plactic relations for two letters a < b, namely, aba = baa, bab = bba, imply the commutativity of noncommutative elementary polynomials in two variables. In other words, the plactic relations for two letters imply that ba(a + b) = (a + b)ba, a < b.
It has been proved in [7] that these relations together with the Knuth relations (1.1) for three letters a < b < c, imply the commutativity of noncommutative elementary symmetric polynomials for any number of variables.
In the present paper we prove that in fact the commutativity of nocommutative elementary symmetric polynomials for n = 2 and n = 3 implies the commutativity of that polynomials for all n, see Theorem 2.23 3 .
One of the main objectives of the present paper is to study combinatorial properties of the generalized plactic Cauchy kernel where P n stands for the set of parameters {p ij , 2 ≤ i + j ≤ n + 1, i > 1, j > 1}, and U := U n stands for a certain noncommutative algebra we are interested in, see Section 5.
We also want to bring to the attention of the reader on some interesting combinatorial properties of rectangular Cauchy kernels F (P n,m , U) = where P n,m = {p ij } 1≤i≤n 1≤j≤m .
We treat these kernels in the (reduced) plactic algebras PC n and PF n,m correspondingly. The algebras PC n and PF n,m are finite dimensional and have bases parameterized by certain Young tableaux described in Section 5.1 and Section 6 correspondingly. Decomposition of the rectangular Cauchy kernel with respect to the basis in the algebra PF n,m mentioned above, gives rise to a set of polynomials which are common generalizations of the (double) Schubert. β-Grothendieck, Demazure and Stanley polynomials. To be more precise, the polynomials listed above correspond to certain quotients of the plactic algebra PF n,m and appropriate specializations of parameters {p ij } involved in our definition of polynomials U α ({p ij }), see Section 6.
As it was pointed out in the beginning of Introduction, the Knuth (or plactic) relations (1.1) have been discovered in [21] in the course of the study of algebraic and combinatorial properties of the Robinson-Schensted correspondence. Motivated by the study of basic properties of a quantum version of the tropical/geometric Robinson-Schensted-Knuth correspondence -work in progress, but see [1], [18], [19], [47], [48] for definition and basic properties of the tropical/geometric RSK, -the author of the present paper came to a discovery that a certain deformations of the Knuth relations preserve the Hilbert series (resp.the Hilbert polynomials) of the plactic algebras P n and F n (resp. the algebras PC n and PF n ).
More precisely, let {q 2 , . . . , q n } be a set of (mutually commuting) parameters, and U n := {u 1 , . . . , u n } be a set of generators of the free associative algebra over Q of rank n. Let Y, Z ⊂ [1, n] be subsets such that Y ∪Z = [1, n] and Y ∩Z = ∅. Let us set p(a) = 0 , if a ∈ Y and p(a) = 1, if a ∈ Z. Define super quantum Knuth relations among the generators u 1 , . . . , u n as follows: SP L q : (−1) p(i)p(k) q k u j u i u k = u j u k u i , i < j ≤ k, (−1) p(i)p(k) q k u i u k u j = u k u i u j , i ≤ j < k.
We define • deformed/quantum superplactic algebra SQP n to be the quotient of the free associative algebra Q u 1 , . . . , u n by the two-sided ideal generated by the set of quantum Knuth relations (SP L q ), • reduced deformed/quantum superplactic algebras SQPC n and SQPF n,m to be the quotient of the algebra SQP n by the two-sided ideals described in Definitions 5.13 and 6.6 correspondingly.
We state Conjecture The algebra SQP n and the algebras SQPC n and SQPF n,m , are flat deformations of the algebras P n , PC n and PF n,m correspondingly.
In fact one can consider more general deformation of the Knuth relations, for example take a set of parameters Q := {q ik , 1 ≤ i < k ≤ n} and impose on the set of generators {u 1 , . . . , u n } the following relations q ik u j u i u k = u j u k u i , i < j ≤ k, q ik u i u k u j = u k u i u j , i ≤ j < k.
However we don't know how to describe a set of conditions on parameters Q which imply the flatness of the corresponding quotient algebra(s), as well as we don't know an interpretation and dimension of the algebras SQPC n and SQPF n,m for a "generic" values of parameters Q. We expect the dimension of algebras SQPC n and SQPF n,m each depends piece-wise polynomially on a set of parameters {q ij ∈ Z ≥0 , 1 ≤ i < j ≤ n} , and pose a problem to describe its polynomiality chambers.
We also mention and leave for a separate publication(s), the case of algebras and polynomials associated with superplactic monoid [44], [27], which corresponds to the relations SP L q with q i = 1, ∀i. Finally we point out on interesting and important paper [43] wherein the case Z = ∅, and the all deformation parameters are equal to each other., has been independently introduced and studied in depth.
Let us repeat that the important property of plactic algebras P n is that the noncommutative elementary polynomials e k (u 1 , . . . , n n−1 ) := n−1≥a 1 ≥a 2 ≥a k ≥1 u a 1 · · · u a k , k = 1, . . . , n − 1, generate a commutative subalgebra inside of the plactic algebra P n , see e.g. [28], [7]. Therefore the all our finite dimensional algebras introduced in the present paper, have a distinguish finite dimensional commutative subalgebra. We have in mined to describe this algebras explicitly in a separate publication.
In Section 2 we state and prove necessary and sufficient conditions in order the elementary noncommutative polynomials form a mutually commuting family. Surprisingly enough to check the commutativity of noncommutative elementary polynomials for any n, it's enough to check these conditions only for n = 2, 3. However a combinatorial meaning of a generalization of the Lascoux-Schützenberger plactic algebra P n invented, is still missing.
The plactic algebra PF n,m introduced in Section 6, has a monomial basis parametrized by the set of Young tableaux of shape λ ⊂ (n m ) filled by the numbers from the set {1, . . . , m}. In the case n = m it is well-known [14], [25], [45], that this number is equal to the number of symmetric plane partitions fit inside the cube n × n × n. Surprisingly enough this number admits a factorization in the product of the number of totally symmetric plane partitions (T SP P ) by the number of totally symmetric self-complementary plane partitions(TSSCPP) fit inside the same cube. A similar phenomenon happens if |m − n| ≤ 2, see Section 6. More precisely,we add to the well-known equalities • #|B 1,n | = 2 n , #|B 2,n | = 2n+1 n , #|B 3,n | = 2 n Cat n , [55], A003645, where AMSHT (2n) denotes the number of alternating sign matrices of size 2n × 2n invariant under a half-turn and CSSP P (2n) denotes the set of cyclically symmetric selfcomplementary plane partitions in the 2n-cube. It is well-known that ASMHT (2n) = ASM(n) × CSP P (n), where CSP P (n) denotes the number of cyclically symmetric plane partitions in n-cube, and CSSCP P (2n) = ASM(n) 2 , see e.g. [3], [26], [55], A006366.
• Construct bijection between the set of plane partitions fit inside n-cube and the set of (ordered) triples (π 1 , π 2 , ℘), where (π 1 , π 2 ) is a pair of T SSCP P (n) and ℘ is a cyclically symmetric plane partition fit inside n-cube.
These relations have strait forward proofs based on the explicit product formulas for the numbers but bijective proofs of these identities are an open problem.
It follows from [28], [38] that the dimension of the (reduced) plactic algebra PC n is equal to the number of alternating sign matrices of size n × n (ASM(n) = T SSCP P (n)). Therefore the Key-Grothendieck polynomials can be obtained from U-polynomials (see Section 6, Theorem 6.9) after the specialization p ij = 0, if i + j > n + 1.
Namely, for any permutation w ∈ S n and composition ζ ⊂ δ n , we introduce polynomials denote a collection of divided difference operators which satisfy the Coxeter and Hecke relations for any reduced decomposition w = s i 1 · · · s i ℓ of a permutation in question; ζ + denotes a unique partition obtained from ζ by ordering its parts, and v ζ ∈ S n denotes the minimal length permutation such that v ζ (ζ) = ζ + . Assume that h = 1 4 . If α = γ = 0, these polynomials coincide with the β-Grothendieck polynomials [8], if β = α = 1, γ = 0 these polynomials coincide with the Di Francesco-Zin-Justin polynomials [12], if β = γ = 0, these polynomials coincide with dual α-Grothendieck polynomials H α w (X).

Conjecture 1.2
For any permutation w ∈ S n and any composition ζ ⊂ δ n , polynomials We expect that these polynomials have some geometrical meaning to be discovered.
More generally we study divided difference type operators of the form depending on parameters a, b, c, h, e and satisfying the 2D-Coxeter relations We find that the necessary and sufficient condition which ensure the validity of the 2D-Coxeter relations is the following relation among the parameters: (a+b)(a-c)+h e = 0 . Therefore, if the above relation between parameters a, b, c, d, h, e hold, the the for any permutation w ∈ S n the operator where w = s i 1 · · · s i ℓ is any reduced decomposition of w, is well-defined. Hence under the same assumption on parameters, for any permutation w ∈ S n one can attach the well-defined polynomial and in much the same fashion to define polynomials for any composition α such that α i ≤ n − i, ∀i. We have used the notation T (x) (a,b,c,d,h,e) w to point out that this operator acts only on the variables X = (x 1 , . . . , x n ); for any composition α ∈ Z n ≥0 , α + denotes a unique partition obtained from α by reordering its parts in (weakly) decreasing order, and w α denotes a unique minimal length permutation in the symmetric group S n such that w α (α) = α + .
In the present paper we are interested in to list a conditions on parameters A := {a, b, c, d, h, e} with the constraint which ensure that the above polynomials G (a,b,c,d,h,e) w (X) and D (a,b,c,d,h,e) α (X) or their specialization x i = 1, ∀i, have nonnegative coefficients. We state the following conjectures: In the present paper we treat the case A = (−β, β + α + γ, γ, 1, (α + γ)(β + γ)).
As it was pointed above, in this case polynomials G A w (X) are common generalization of Schubert,β-Grothendieck and dual β-Grothendieck, and Di Francesco-Zin-Justin polynomials. We expect a certain c interpretation of the polynomials G A w for general β, α and γ. As it was pointed out earlier, one of the basic properties of the plactic monoid P n is that the nonocommutative elementary symmetric polynomials {e k (u 1 , . . . , u n−1 )} 1≤k≤n−1 generate a commutative subalgebra in the plactic algebra in question. One can reformulate this statement as follows. Consider the generating function where we set e 0 (U) = 1. Then the commutativity property of noncommutative elementary symmetric polynomials is equivalent to the following commutativity relation in the plactic as well as in the generic plactic, algebras P n and P n , [7], and Theorem 2.23,

Now let us consider the Cauchy kernel
where we assume that the pairwise commuting variables z 1 , . . . , z n−1 commute with the all generators of the algebras P n and P n . In what follows we consider the natural completion P n of the plactic algebra P n to allow consider elements of the form (1 + x u i ) −1 . Elements of this form exist in any Hecke type quotient of the plactic algebra P n . Having in mind this assumption, let us compute the action of divided difference operators ∂ z i,i+1 on the Cauchy kernel. In the computation below, the commutativity property of the elements A i (x) and A i (y) plays the key role. Let us start computation of According to the basic property of the elements A i (x), one sees that the expression A i (z i ) A i (z i+1 ) is symmetric with respect to z i and z i+1 , and hence is invariant under the action of divided difference operator ∂ z i,i+1 Therefore.

It is clearly seen that
It is easy to see that if one adds Hecke's type relations on the generators .
Therefore in the quotient of the plactic algebra P n by the Hecke type relations listed above and by the "locality" relations one obtains Finally, if a = 0, then the above identity takes the following form In other words the above identity is equivalent to the statement [9] that in the IdCoxeter algebra IC n the Cauchy kernel C(P n , U) is the generating function for the b-Grothendieck polynomials. Moire over, each (generalized) double b-Grothendieck polynomial is a positive linear combination of the key-Grothendieck polynomials. In the special case b = −1 and P ij = x i + y j if 2 ≤ i + j ≤ n + 1, p ij = 0, if i + j > n + 1 this result had been stated in [39].
As a possible mean to define affine versions of polynomials treated in the present paper, we introduce the double affine nilCoxeter algebra of type A and give construction of a generic family of Hecke's type elements 5 we will be put to use in the present paper.
As Appendix we include several examples of polynomials studied in the present paper to illustrate results obtained in these notes. We also include an expository text concerning the MacNeille completion of a poset to draw attention of the reader to this subject. It is the MacNeille completion of the poset associated with the (strong) Bruhat order on the symmetric group, that was one of the main streams of the study in the present paper.
A bit of history. Originally these notes have been designed as a continuation of [8]. The main purpose was to extend the methods developed in [10] to obtain by the use of plactic algebra, a noncommutative generating function for the key (or Demazure) polynomials introduced by A. Lascoux and M.-P. Schützenberger [34]. The results concerning the polynomials introduced in Section 4, except the Hecke-Grothendieck polynomials, see Definition 4.6, has been presented in my lecture-courses "Schubert Calculus" have been delivered in the Graduate School of Mathematical Sciences, the University of Tokyo, November 1995 -April 1996, and in the Graduate School of Mathematics, Nagoya University, October 1998 -April 1999. I want to thank Professor M. Noumi and Professor T. Nakanishi who made these courses possible. Some early versions of the present notes are circulated around the world and now I was asked to put it for the wide audience. I would like to thank Professor M. Ishikawa (Department of Mathematics, Faculty of Education, University of the Ryukyus, Okinawa, Japan) and Professor S.Okada (Graduate School of Mathematics, Nagoya University, Nagoya, Japan) for valuable comments.  The plactic algebra P n is an (unital) associative algebra over Z generated by elements {u 1 , · · · , u n−1 } subject to the set of relations Proposition 2.2 ( [28])Tableau words in the alphabet U = {u 1 , · · · , u n−1 } form a basis in the plactic algebra P n .
In other words, each plactic class contain a unique tableau word. In particular,

Remark 2.3
There exists another algebra over Z which has the same Hilbert series as that of the plactic algebra P n . Namely, define algebra L n to be an associative algebra over Z generated by the elements {e 1 , e 2 , . . . , e n−1 }, subject to the set of relations (e i , (e j , e k )) := e i e j e k − e j e i e k − e j e k e i + e k e j e i = 0, f or all 1 ≤ i, j, k ≤ n − 1, j < k.
Note that the number of defining relations in the algebra L n is equal to 2 n 3 . One can show that the dimension of the degree k homogeneous component L (k) n of the algebra L n is equal to the number semistandard Young tableaux of the size k filled by the numbers from the set {1, 2, . . . , n}.

Definition 2.4
The local plactic algebra LP n is an associative algebra over Z generated by elements {u 1 , . . . , u n−1 } subject to the set of relations One can show (A.K) that

Definition 2.5 (Nil Temperley-Lieb algebra)
Denote by T L (0) n the quotient of the local plactic algebra LP n by the two-sided ideal generated by the elements {u 2 1 , . . . , u 2 n−1 }.

Proposition 2.6
The Hilbert polynomial Hilb(T L (0) n , t) is equal to the generating function for the number of 321-avoiding permutations of the set {1, 2, ..., n} having inversion number equal to k, see [55], A140717, for other combinatorial interpretations of polynomials Hilb(T L (0) n , t).
We denote by T L (β) n the quotient of the local plactic algebra LP n by the two-sided ideal generated by the elements {u 2 1 − β u 1 , . . . , u 2 n−1 − β u n−1 }.

Definition 2.7
The modified plactic algebra MP n is an associative algebra over Z generated by {u 1 , . . . , u n−1 } subject to the set of relations (P L1) and that Definition 2.8 The (reduced) nilplactic algebra N P n is an associative algebra over Q generated by {u 1 , · · · , u n−1 } subject to the relations 6 the set of relations (P L1), and that Proposition 2.10 The nilplactic algebra N P n has finite dimension, its Hilbert polynomial Hilb(N P n , t) has degree n 2 and dim(N P n ) ( n 2 ) = 1.

Definition 2.12
The idplactic algebra IP (β) n is an associative algebra over Q(β) generated by {u 1 , · · · , u n−1 } subject to the relations 5) and the set of relations (P L1).
In other words, the idplactic algebra IP n is the quotient of the plactic algebra P n by the the two-sided ideal generated by elements Proposition 2.13 Each idlplactic class contains a unique tableau word of the smallest length.
For each word w denote by rl(w) the length of a unique tableau word of minimal length which is idplactic equivalent to w. 6 Original definition of the nilplactic relations given in [32] involves only relations (P L1) and It had been shown [33] that the Schensted construction for the plactic congruence extends to the nilplactic case. However as it seen from the following example, as a consequence of relations (P L1) one has and therefore noncommutative elementary symmetric polynomials e 1 (u 1 , u 2 , u 3 ) and e 2 ((u 1 , u 2 , u 3 ) do not commute modulo the nilplactic congruence defined in [32]. Indeed, u 1 u 3 u 1 ≡ u 3 u 1 u 3 . In order to guarantee the commutativity of all noncommutative elementary polynomials, we add relations Cf with definition of idplactic relations listed in Definition 2.11. Example 2.14 Consider words in the alphabet {a < b < c < d}. Then rl(dbadc) = 4 = rl(cadbd), rl(dbadbc) = 5 = rl(cbadbd). Indeed, Note that according to our definition, tableau words w = 31, w = 13 and w = 313 belong to different idplactic classes.

Proposition 2.15
The idplactic algebra IP (β) n has finite dimension, and its Hilbert polynomial has degree n 2 .

Definition 2.17
The idplactic Temperly-Lieb algebra PT L (β) n is define to be the quotient of the idplactic algebra IP (β) n by the two-sided ideal generated by the elements   , and Coef f t max Hilb(PT L n , t) = 1, if n is even, and = 2, if n is odd.

Definition 2.18
The nilCoxeter algebra N C n is defined to be the quotient of the nilplactic algebra N P n by the two-sided ideal generated by elements Clearly the nilCoxeter algebra N C n is a quotient of the modified plactic algebra MP n by the two-sided ideal generated by the elements

Definition 2.19
The idCoxeter algebra IC (β) n is defined to be the quotient of the idplactic algebra IP (β) n by the two-sided ideal generated by the elements It is well-known that the algebras N C n and IC (β) n have dimension n!, and the elements {u w := u i 1 · · · u i ℓ }, where w = s i 1 · · · s i ℓ is any reduced decomposition of w ∈ S n , form a basis in the nilCoxeter and idCoxeter algebras N C n and IC (β) n .

Remark 2.20
There is a common generalization of the algebras defined above which is due to S.Fomin and C.Greene [7]. Namely, define generalized plactic algebra P n to be an associative algebra generated by elements u 1 , · · · , u n−1 , subject to the relations (P L2) and relations The relation (2.5) can be written also in the form Then the elements A i,j (x) and A i,j (y) commute in the generalized plactic algebra P n .
Moreover, the algebra C 1,n is a maximal commutative subalgebra of P n .
To establish Theorem 2.20 , we are going to prove more general result. To start with, let us define generic plactic algebra P n .

Definition 2.23
The generic plactic algebra P n is an associative algebra over Z generated by {e 1 , · · · , e n−1 } subject to the set of relations Clearly seen that relations (2.6)−(2.8) are consequence of the plactic relations (P L1) and (P L2).

Theorem 2.24 Define
Then the elements A n (x) and A n (y) commute in the generic plactic algebra P n . Moreover the elements A n (x) and A n (y) commute if and only if the generators {e 1 , . . . , e n−1 } satisfy the relations (2.6) − (2.8).
Proof For n = 2, 3 the statement of Theorem 1.22 is obvious. Now assume that the statement of Theorem 1.22 is true in the algebra P n . We have to prove that the commuta- Using relations (2.7) we can move the commutator (e i , e n ) to the left, since i < a < n, till we meet the term (1 + xe n ). Using relations (2.6) we see that Therefore we come to the following relation Finally let us observe that Finally, if i < j, then (e i + e j , e j e i ) = 0 ⇐⇒ (2.6), if i < j < k and the relations (2.6) hold, then (e i + e j + e k , e j e i + e k e j + e k e i ) = 0 ⇐⇒ (2.7), if i < j < k and relations (2.6) and (2.7) hold, then (e i + e j + e k , e k e j e i ) = 0 ⇐⇒ (2.8); the relations (e j e i + e k e j + e k e i , e k e j e i ) = 0 are a consequence of the above ones.
Definition 2.25 (Compatible sequences b) Given a word a ∈ R(T ) (resp. a ∈ IR(T )), denote by C(a) (resp. IC(a)) the set of sequences of positive integers, called compatible sequences, b : (2.10) Finally, define the set C(T ) (resp. IC(T )) to be the union C(a) (resp. the union IC(a)), where a runs over all words which are plactic (resp. idplactic) equivalent to the word w(T ).
be the set of (mutually commuting) variables.
Definition 2.27 (1) Let T be a semistandard tableau, and n := |T |. Define the double key polynomial K T (P) corresponding to the tableau T to be (2.11) (2) Let T be a semistandard tableau, and n := |T |. Define the double key Grothendieck polynomial GK T (P) corresponding to the tableau T to be (2.12) In the case when p i,j = x i + y j , ∀i, j, where X = {x 1 , . . . , x n } and Y = {y 1 , . . . , y n } denote two sets of variables, we will write K T (X, Y ), GK T (X, Y ), . . . , instead of K T (P), GK T (P), . . . .

Definition 2.28
Let T be a semistandard tableau, denote by α(T ) = (α 1 , · · · , α n ) the exponent of the smallest monomial in the set We will call the composition α(T ) to be the bottom code of tableau T.

Divided difference operators
In this subsection we remind some basic properties of divided difference operators will be put to use in subsequent Sections. For more details, see [46]. Let f be a function of the variables x and y (and possibly other variables), and η = 0 be a parameter. Define the divided difference operator ∂ xy (η) will as follows where the operator s η xy acts on the variables (x, y, . . .) according to the rule: s η xy transforms the pair (x, y) to (η −1 y, η x), and fixes all other variables. We set by definition, s η yx := s η −1 xy . The operator ∂ xy (η) takes polynomials to polynomials and has degree −1. The case η = 1 corresponds to the Newton divided difference operator ∂ xy := ∂ xy (1).
Let x 1 , . . . , x n be independent variables, and let P n := Q[x 1 , . . . , x n ]. For each i < j put It is interesting to consider also an additive or affine analog ∂ xy [k] of the divided difference operators ∂ xy (η), namely,

Remark 4.2
We can also introduce polynomials Z w , which are defined recursively as follows: However, one can show that , formula (6), had been used by A.Lascoux to describe the transition on Grothendieck polynomials, i.e. stable decomposition of any Grothendieck polynomial corresponding to a permutation w ∈ S n . into a sum of Grasmannian ones corresponding to a collection of Grasmannin permutations v λ ∈ S ∞ , see [37] for details. The above mentioned operators D i had been used in [37] to construct a basis Ω α | α ∈ Z ≥0 that deforms the basis which is built up from the Demazure ( known also as key) polynomials. Therefore polynomials KG[α](X; β = −1) coincide with those introduced by A. Lascoux in [37]. In [51] the authors give a conjectural construction for polynomials Ω α based on the use of extended Kohnert moves, see e.g. [45], Appendix by N. Bergeron, for definition of the Kohnert moves. We state Conjecture that α are defined in [51] using the K-theoretic versions of the Kohnert moves. For β = −1 this Conjecture has been stated in [51]. It seems an interesting problem to relate the K-theoretic Kohnert moves with certain moves of 1 ′ s introduced in [8].
We will use notation S w (X), G w (X), ..., for polynomials S w (X, 0), G w (X, 0), ... . • Di Francesco-Zin-Justin polynomials) Definition 4.4 For each permutation w ∈ S n the Di Francesco-Zinn-Justin polynomials DZ w (X) are defined recursively as follows: if w is the longest element in S n , then DZ w (X) = R δ (X, 0); otherwise, if w and i are such that w i > w i+1 , i.e. l(ws i ) = l(w) − 1, then (1) Polynomials DZ w (X) have nonnegative integer coefficients.
(2) For each permutation w ∈ S n the polynomial DZ w (X) is a linear combination of key polynomials K[α](X) with nonnegative integer coefficients.
As for definition of the double Di Francesco-Zin-Justin polynomials DZ w (X, Y ) they are well defined, but may have negative coefficients.

Definition 4.6
Let w ∈ S n , define Hecke-Grothendieck polynomials KN β,α w (X n ) to be where as before x δn := x n−1 where u = s i 1 · · · s i ℓ is any reduced decomposition of a permutation taken.
• More generally, let β, α and γ be parameters, consider divided difference operators For a permutation w ∈ S n define polynomials where w = s i 1 · · · s i ℓ is any reduced decomposition of w.

Remark 4.7
A few comments in order. (a) The divided difference operators {T i := T (β,α,γ) i 1 , i = 1, · · · , n − 1} satisfy the following relations • (Hecke relations) Therefore the elements T β,α w are well defined for any w ∈ S n . • (Inversion) w constitute a common generalization of the β-Grothendieck polynomials , namely, G • (Stability) Let w ∈ S n be a permutation and w = s i 1 s i 2 · · · s i ℓ be any its reduced decomposition. Assume that i a ≤ n − 3, ∀ 1 ≤ a ≤ ℓ, and define permutation w := (1) is equal to the degree of the variety of pairs commuting matrices of size n × n, • the bidegree of the affine homogeneous variety V w , w ∈ S n , [12], is equal to .
Note that the assmption β = 0 is necessary.
The number KN (β=1,α=1) w (1) is equal to the number of Schröder paths of semilength (n-1) in which the (2, 0)-steps come in 3 colors and with no peaks at level 1, see [55], A162326 for further properties of these numbers.
It is well-known, see e.g. [55], A126216, that the polynomial KN (β,α=0) w (1) counts the number of dissections of a convex (n + 1)-gon according the number of diagonals involved, where as the polynomial KN (β,α) w (1) (up to a normalization) is equal to the bidegree of certain algebraic varieties introduced and studied by A. Knutson [22].
A few comments in order.  We state more general Conjecture in Introduction. In the present paper we treat only the case r = 0, since a combinatorial meaning of polynomials KN (a,b,c,a+c+r) w (1) in the the case r = 0 is missed for the author.  Let α = (α 1 ≤ α 2 ≤ · · · ≤ α r ) be a composition, define partition α + = (α r ≥ · · · ≥ α 1 ).

Proposition 4.12
If α = (α 1 ≤ α 2 ≤ · · · ≤ α r ) is a composition and n ≥ r, then For example, KG[0, 1, 2, · · · , n − 1] = 1≤i<j≤n (x i + x j + x i x j ). Note that Comments 4.1 Definition 4.13 Define degenerate affine 2d nil-Coxeter algebra AN C (2) n to be an associative algebra over Q generated by the set of elements {{u i,j } 1≤i<j≤n and x 1 , . . . , x n } subject to the set of relations Now for a set of parameters 8 A := (a, b, c, h, e) define elements are valid, if and only if the following relation among parameters a, b, c, e, h holds 9 (a + b)(a − c) + h e = 0. (4.13) (3) (Yang-Baxter relations) Relations 8 By definition, a parameter assumed to be belongs to the center of the algebra in question 9 The relation (4.13) between parameters a, b, c, e, h defines a rational four dimensional hypersurface. Its open chart {e h = 0} contains, for example, the following set (cf [37]): {a = p 1 p 4 − p 2 p 3 , b = p 2 p 3 , c = p 1 p 4 , e = p 1 p 3 , h = p 2 p 4 }, where (p 1 , p 2 , p 3 , p 4 ) are arbitrary parameters. However the points (−b, a + b + c, c, 1, (a + c)(b + c), (a, b, c) ∈ N 3 } do not belong to this set (5) Assume that parameters a, b, c, h, e satisfy the conditions (4.13) and that b c+1 = h e.

Example 4.16
• Each of the set of elements by itself generate the symmetric group S n .
• If one adds the affine elements s • It seems an interesting problem to classify all rational, trigonometric and elliptic divided difference operators satisfying the Coxeter relations. A general divided difference operator with polynomial coefficients had been constructed in [31], see also Lemma 4.14,(4.13). One can construct a family of rational representations of the symmetric group (as well as its affine extension) by "iterating" the transformations s  A = (a, b, c, h, e) be a sequence of integers satisfying the conditions (4.5). Denote by ∂ A i the divided difference operator

Definition 4.17
(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.

Lemma 4.18
Let w ∈ S n be a permutation. • If A = (0, 0, 0, 1, 0), then S A w (X n ) is equal to the Schubert polynomial S w (X n ).
In all cases listed above the polynomials S A w (X n ) have non-negative integer coefficients. . Define the generalized key or Demazure polynomial corresponding to a composition α as follows 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. .

•
If A = (−1, q −1 , −1, 0, 0) and λ is a partition, then (up to a scalar factor) polynomial K A λ (X n ) can be identify with a certain Whittaker function (of type A), see [4], Theorem A. Note that operator

satisfy the Coxeter and Hecke relations, namely (T
In [4] the operator T A i has been denoted by T i . • Let w ∈ S n be a permutation and m = (i 1 , . . . , i ℓ ) be a reduced word for w, i.e. w = s i 1 · · · s i ℓ and ℓ(w) = ℓ. Denote by Z m the Bott-Samelson nonsingular variety corresponding to the reduce word m. It is well-known that the Bott-Samelson variety Z m is birationally isomorphic to the Schubert variety X w associated with permutation w, i.e. the Bott-Samelson variety Z m is a desingularization of the Schubert variety X w . Follow [4] define the Bott-Samelson polynomials Z m (x, λ, v) as follows • If A = (−β, β + α, 0, 1, βα) , then S A w (X n ) constitutes a common generalization of the Grothendieck and the Di Francesco-Zin-Justin polynomials.
It is easily seen that φ T i = T i+1 φ, i = 0, · · · , n − 2, and φ 2 T n−1 = T 1 φ 2 . It has been established in [41] how to use the operators φ, T 1 , . . . , T n−1 to to give formulas for the interpolation Macdonald polynomials. Using operators φ, T where T i = T t,−t,1,0,0 i , generate a commutative subalgebra in the double affine nilCoxeter algebra DANC n . Note that the algebra DANC n contains lot of other interesting commutative subalgebras, see e.g. [16].
It seems interesting to give an interpretation of polynomials generated by the set of operators T t,−t,1,h,e i , i = 0, · · · , n − 1 in a way similar to that given in [41]. We expect that these polynomials provide an affine version of polynomials KN (−t,−1,1,1,0) w (X), w ∈ S n ⊂ S af f n , see Remark 4.7.
Note that for any affine permutation v ∈ S af f n , the operator where v = s i 1 · · · s i ℓ is any reduced decomposition of v, is well-defined up to the sign ±1. It seems an interesting problem to investigate properties of polynomials L v [α](X n ), where v ∈ S af f n and α ∈ Z n ≥0 , and find its algebra-geometric interpretations.

Cauchy kernel
Let u 1 , u 2 , · · · , u n−1 be a set of generators of the free algebra F n−1 , which assumed also to be commute with the all variables P n := {p i,j , 2 ≤ i + j ≤ n + 1, i ≥ 1, j ≥ 1}..

Definition 5.1
The Cauchy kernel C(P n , U) is defined to be as the ordered product (5.14) For example, In the case {p ij = x i , ∀j} we will write C n (X, U) instead of C(P n , U).
where a = (a 1 , . . . , a p ), b = (b 1 , . . . , b p ), w(a, b) = p j=1 u a j +b j −1 , and the sum in (4.10) runs over the set S n := We denote by S (0) n the set {(a, b) ∈ S n | w(a, b) is a tableau word}. The number of terms in the right hand side of (5.15) is equal to 2 ( n 2 ) , and therefore is equal to the number #|ST Y (δ n , ≤ n)| of semistandard Young tableaux of the staircase shape δ n := (n−1, n−2, . . . , 2, 1) filled by the numbers from the set {1, 2, . . . , n}. It is also easily seen that the all terms appearing in the RHS(4.10) are different, and thus #|S n | = #|ST Y (δ n , ≤ n)|.
We are interested in the decompositions of the Cauchy kernel C(P n , U) in the algebras P n , N P n , IP n , N C n and IC n .

Plactic algebra P n
Let λ be a partition and α be a composition of the same size. Denote by ST Y (λ, α) the set of semistandard Young tableaux T of the shape λ and content α which must satisfy the following conditions: • for each k = 1, 2, · · · , the all numbers k are located in the first k columns of the tableau T . In other words, the all entries T (i, j) of a semistandard tableau T ∈ ST Y (λ, α) have to satisfy the following conditions: T i,j ≤ j. For a given (semi-standard) Young tableau T let us denote by R i (T ) the set of numbers placed in the i-th row of T , and denote by ST Y 0 (λ, α) the subset of the set ST Y 0 (λ, α) involving only tableaux T which satisfy the following constrains : To continue, let us denote by A n (respectively by A (0) n ) the union of the sets ST Y (λ, α) (resp. that of ST Y 0 (λ, α)) for all partitions λ such that λ i ≤ n − i for i = 1, 2, · · · , n − 1, and all compositions α, l(α) ≤ n − 1. Finally, denote by A n (λ) (resp.A (0) n (λ)) the subset of A n (resp. A (0) n (λ)) consisting of all tableaux of the shape λ.
• There exists a bijection ρ n : A n −→ ASM(n) such that the image Im (A (0) n ) contains the set of n × n permutation matrices.
• The number of column strict, as well as row strict diagrams which are contained inside the staircase diagram (n, n − 1, . . . , 2, 1) is equal to 2 n . We expect that the image ρ n ( n−1 k=0 A n ((k))) coincides with the set of n × n permutation matrices corresponding to either 321-avoiding or 132-avoiding permutations. Now we are going to define a statistic n(T ) on the set A n .
Definition 5.5 Let λ be a partition, α be a composition of the same size. For each tableau It is instructive to display the numbers {A n (λ), λ ⊂ δ n } as a vector of the length equals to the n − th Catalan number. For example, It is easy to see that the above data, as well as the corresponding data for n = 5, coincide with the list of refined totally symmetric self-complementary plane partitions that fit in the box 2n × 2n × 2n (T SSCP P (n) for short) listed for n = 1, 2, 3, 4, 5 in [12], Appendix D.
In particular, λ⊂δn A λ (t) = 1≤j≤n−1 A n,j t j−1 , where A n,j stands for the number of alternating sign matrices (ASM n for short) of size n × n with a 1 on top of the j-th column. |A n | = |T SSCP P (n)| = |ASM n |.

It is well-known [3] that
and the total number A n of ASM of size n × n is equal to where F n denotes the number of forests of trees on n labeled nodes; K ρn,λ denotes the Kostka number, i.e. the number of semistandard Young tableaux of the shape ρ n := (n − 1, n − 2, . . . , 1) and content/weight λ; for any partition λ = (λ 1 ≥ λ 2 ≥ . . . ≥ λ n ≥ 0) we set m i (λ) = {j | λ j = i}.
Note that the rigged configuration bijection gives rise to an embedding of the set of labeled regular tournaments with n := 2k + 1 nodes to the set ST Y (ρ n , ≤ n), if n is an odd integer, and to the set ST Y (ρ n−1 , ≤ n − 1), if n is even.

Definition 5.13
Define algebra PC n to be the quotient of the plactic algebra P n by the twosided ideal J n by the set of monomials Theorem 5.14 • The algebra PC n has dimension equals to ASM(n), • Hilb(PC n , q) = λ∈δ n−1 |A λ | q |λ| , • Hilb((PC n+1 ) ab , q) = n k=0 n−k+1 n+1 n+k n q k , cf [55], A009766. 10 For the reader convenience we recall a definition of a tableau word. Let T be a (regular shape) semistandard Young tableau. The tableau word w(T ) associated with T is the reading word of T is the sequence of entries of T obtained by concatenating the columns of T bottom to top consecutively starting from the first column. For example, take .
The corresponding tableau word is w(T ) = 5321432433. By definition, a tableau word is the tableau word corresponding to some (regular shape) semistandard Young tableau. It is well-known [34] that the number of tableau subwords contained in I 0 is equal to the number of alternating sign matrices ASM (n).

Definition 5.15
Denote by PC ♯ n the quotient of the algebra PC n by the two-sided ideal generated by the elements {u i u j − u j u i , |i − j| ≥ 2}.
Proposition 5.16 Dimension dim PC ♯ n of the algebra PC ♯ n is equal to the number of Dyck paths whose ascent lengths are exactly {1, 2, . . . , n + 1}.
See [55],A107877 where the first few of these numbers are displayed.

Problem 5.19
Denote by A n the algebra generated by the curvature of 2-forms of the tautological Hermitian linear bundles ξ i , 1 ≤ i ≤ n, over the flag variety F l n , [54]. It is well-known [50] that the Hilbert polynomial of the algebra A n is equal to where the sum runs over the set F (n) of forests F on the n labeled vertices, and inv(F ) (resp. maj(F )) denotes the inversion index (resp. the major index) of a forest F. 11 Clearly that dim(A n ) ( n 2 ) ) = dim(PC n ) ( n 2 ) = dim(H ⋆ (F l n , Q) ( n 2 ) = 1. = s(n + 2, 2), where s(n, k) demotes the Stirling number of the first kind, see e.g. [55], A000914. 11 For the readers convenience we recall definitions of statistics inv(F ) and maj(F ). Given a forest F on n labeled vertices, one can construct a tree T by adding a new vertex (root) connected with the maximal vertices in the connected components of F.
The inversion index inv(F ) is equal to the number of pairs (i, j) such that 1 ≤ i < j ≤ n, and the vertex labeled by j lies on the shortest path in T from the vertex labeled by i to the root.
The major index maj(F ) is equal to x∈Des(F ) h(x); here for any vertex x ∈ F , h(x) is the size of the subtree rooted at x; the descent set Des(F ) of F consists of the vertices x ∈ F which have the labeling strictly greater than the labeling of its child.

Problems (1) Is it true that
Hilb(PC n , t) − Hilb(A n , t) ∈ N[t] ? If so, as we expect, does there exist an embedding of sets ι : F (n) ֒→ A n such that inv(F ) = n(ι(F )) for all F ∈ F n ? See Section 5.1 for definitions of the set A n and statistics n(T ), T ∈ A n , Definition 5.5.

Comments 5.3
One can ask a natural question : when does noncommutative elementary polynomials e 1 (A), · · · , e n (A) form a q-commuting family, i.e. e i (A) e j (A) = q e j (A) e i (A), 1 ≤ i < j ≤ n ?
Clearly that in the case of two variables one needs to necessitate the following relations e i e j e i + e j e j e i = q e j e i e i + q e j e i e j , i < j.
Having in mind to construct a quantization, or q-analogue of the plactic algebra P n , one would be forced to the following relations q e j e i e j = e j e j e i and q e j e i e i = e i e i e j e i , i < j.
It is easily seen that these two relation are compatible iff q 2 = 1. Indeed.
e j e j e i e j = q e j e i e j e i = q 2 e j e j e i e i , =⇒ q 2 = 1.
In the case q = 1 one comes to the Knuth relations (P L 1) and (P L 2). In the case q = −1 one comes to the "odd" analogue of the Knuth relations, or "odd" plactic relations (OP L n ), i.e., (OP L n ) : More generally, let Q n := {q ij } 1≤i<j≤n−1 be a set of parameters. Define generalized plactic algebra QP n to be (unital) associative algebra over the ring Z[{q ±1 ij } 1≤i<j≤n−1 ] generated by elements u 1 , . . . , u n−1 subject to the set of relations Proposition 5.21 Assume that q ij := q j , ∀ 1 ≤ i < j. Then the reduced generalized plactic algebra QPC n is a free Z[q ±1 2 , . . . , q ±1 n−1 ]-module of rank equals to the number of alternating sign matrices ASM(n). Moreover, Hilb(QPC n , t) = Hilb(PC n , t), Hilb(QP n , t) = Hilb(P n , t).
Recall that reduced generalized plactic algebra is the quotient of the generalized plactic algebra by the two-sided ideal J n introduced in Definition 5.13.

Example 5.22
(A) (Super plactic monoid, [44], [27]) Assume that the set of generators U := {u 1 , . . . , u n−1 } is divided on two non-crossing subsets, say Y and Z, Y ∪ Z = U, Y ∩ Z = ∅. To each element u ∈ U let us assign the weight wt(u) as follows: wt(u) = 0, if u ∈ Y , and wt(u) = 1 if u ∈ Z. Finally, define parameters of the generalized plactic algebra QP n to be q ij = (−1) wt(u i ) wt(u j ) . As a result we led to conclude that the generalized plactic algebra QP n in question coincides with the super plactic algebra PS(V ) introduced in [44]. We will denote this algebra by SP k,l , where k = |Y |, l = |Z|. We refer the reader to papers [44] and [27] for more details about connection of the super plactic algebra and super Young tableaux, and super analogue of the Robinson-Schensted -Knuth correspondence. We are planning to report on some properties of the Cauchy kernel in the super plactic algebra elsewhere.
(B) (q-analogue of plactic algebra) Now let q = 0, ±1 be a parameter, and assume that q ij = q, ∀ 1 ≤ i < j ≤ n − 1. This case has been treated recently in [43]. We expect that the generalized Knuth relations (5.17) are related with quantum version of the tropical/geometric RSK-correspondence (work in progress), and, probably, with a q-weighted version of the Robinson-Schensted algorithm, presented in [48]. Another interesting problem is to understand a meaning of Q-plactic polynomials coming from the decomposition of the Cauchy kernels C n and F n in the reduced generalized plactic algebra QPC n .

Nilplactic algebra N P n
Let λ be a partition and α be a composition of the same size. Denote by ST Y (λ, α) the set of columns and rows strict Young tableaux T of the shape λ and content α such that the corresponding tableau word w(T ) is reduced, i.e. l(w(T )) = |T |.

Theorem 5.23
(1) In the nilplactic algebra N P n the Cauchy kernel has the following decomposition (2) Let T ∈ B n be a tableau, and assume that its bottom code is a partition. Then Example 5.24 For n = 4 one has C 4 (X, U) =

Idplactic algebra IP n
Let λ be a partition and α be a composition of the same size. Denote by ST Y (λ, α) the set of columns and rows strict Young tableaux T of the shape λ and content α such that l(w(T )) = rl(w(T )), i.e. the tableau word w(T ) is a unique tableau word of minimal length in the idplactic class of w(T ), cf Example 1.9. Denote by D n the union of the sets ST Y (λ, α) for all partitions λ such that λ i ≤ n − i for i = 1, 2, · · · , n − 1, and all compositions α, l(α) ≤ n − 1.

Theorem 5.25
In the idplactic algebra IP n the Cauchy kernel has the following decomposition (2) Let T ∈ D n be a tableau, and assume that its bottom code is a partition. Then

NilCoxeter algebra N C n
Theorem 5.28 In the nilCoxeter algebra N C n the Cauchy kernel has the following decomposition Let w ∈ S n be a permutation, denote by R(w) the set of all its reduced decompositions. Since the nilCoxeter algebra N C n is the quotient of the nilplactic algebra N P n , the set R(w) is the union of nilplactic classes of some tableau words w(T i ) : R(w) = C(T i ). Moreover, R(w) consists of only one nilplactic class if and only if w is a vexillary permutation. In general case we see that the set of compatible sequences CR(w) for permutation w is the union of sets C(T i ).

Corollary 5.29
Let w ∈ S n be a permutation of length l, then (2) Double Schubert polynomial S w (X, Y ) is a linear combination of double key polynomials K T (X, Y ), T ∈ B n , w = w(T ), with nonnegative integer coefficients.

IdCoxeter algebras
A few remarks in order.
Let w ∈ S n be a permutation, denote by IR(w) the set of all decompositions in the idCoxeter algebra IC n of the element u w as the product of the generators u i , 1 ≤ i ≤ n − 1, of the algebra IC n . Since the idCoxeter algebra IC n is the quotient of the idplactic algebra IP n , the set IR(w) is the union of idplactic classes of some tableau words w(T i ) : IR(w) = IR(T i ). Moreover, the set of compatible sequences IC(w) for permutation w is the union of sets IC(T i ).
Corollary 5.32 Let w ∈ S n be a permutation of length l, then is a linear combination of double key Grothendieck polynomials KG T (X, Y ), T ∈ B n , w = w(T ), with nonnegative integer coefficients.

F-kernel and symmetric plane partitions
Let us fix natural number n and k, and a partition λ ⊂ (n k ). Clearly the number of such partitions is equal to n+k n ; note that in the case n = k the number 2n n is equal to the Catalan number of type B n .
Denote by B n,k (λ) the set of semistandard Young tableaux of shape λ filled by the numbers from the set {1, 2, . . . , n}. For a tableau T ∈ B n,k set as before, Denote by B n,k := λ⊂(n k ) B n,k (λ).
Lemma 6.1 ( [14], [25] ) The number of elements in the set B n,k is equal to See also [55], A073165 for other combinatorial interpretations of the numbers #|B n,k |. For example, the number #|B n,k | is equal to the number of symmetric plane partitions fit inside the box n × k × k.
Note that in the case n = k the number B n := B n,n is equal to the number of symmetric plane portions fit inside the n × n × n-box, see [55], A049505. Let us point to that in general it may happen that the number #|B n,n+2 | does not divisible by any ASM(m), m ≥ 3. For example, B 3,5 = 4224 = 2 5 × 3 × 11. On the other hand, it's possible that the number #|B n,n+2 | is divisible by ASM(n = 1), but does not divisible by ASM(n + 2). For example, B 4,6 = 306735 = 715 × 429, but 306735 ∤ 7436 = ASM(6).

Problem 6.8
Let Γ := Γ n,m k,ℓ = (n k , m ℓ ), n ≥ m be a "fat hook". Find generalizations of the identity (6.21) and those listed in [17], p. 71, to the case of fat hooks, namely to find "nice" expressions for the following sums • Find "bosonic" type formulas for these sum at the limit n −→ ∞, ℓ −→ ∞, m, k are fixed.
• (Plactic decomposition of the F n -kernel) where summation runs over the set of semistandard Young tableaux T of shape λ ⊂ (n) m filled by the numbers from the set {1, . . . , m}. • , where λ denotes the shape of a tableau T , and λ ′ denotes the conjugate/transpose of a partition λ.
• The polynomial L n (d) has non-negative coefficients, and polynomial L n (d) + d n is symmetric and unimodal.

MacMeille completion of a partially ordered set 12
Let (Σ, ≤) be a partially ordered set (poset for short) and X ⊆ Σ. Define • The set of upper bounds for X, namely, • The set of lower bounds for X, namely,  In the present paper we are interesting in properties of the MacNeille completion of the Bruhat poset B n = B(S n ) corresponding to the symmetric group S n . Below we briefly describe a construction of the MacNeille completion L n (S n ) := MN n (B n ) follow [28], and [57], v. 2, p. 552, d.

Theorem 7.8 ([28])
The poset L ( S n ) is a complete distributive lattice with number of vertices equals to the number ASM(n) that is the number of alternating sigh matrices of size n × n. Moreover, the lattice L ( S n ) is order isomorphic to the MacNeille completion of the Bruhat poset B n .
Indeed it is not difficult to prove that the set of all monotonic triangles obtained by applying repeatedly operation (=meet) to the set {T (w), w ∈ S n of triangles corresponding to all elements of the symmetric group S n , coincides with the set of all monotonic triangles L(S ⋉ . The natural map κ : S n −→ L(S n is obviously embedding, and all other conditions of Proposition 7.2 are satisfied. Therefore L(S n ) = MN (B n .the fact that the lattice L(S n is a distributive one follows from the well-known identity max(x, min(y, z)) = min(max(x, y), max(x, z)), x, y, z ∈ R ≥0 ) 3 .
Finally the fact that the cardinality of the lattice L(S n ) is equal to the number ASM(n) had been proved by A. Lascoux and M.-P. Schützenberger [28].
If T = [t ij ] ∈ L(S n ), define rank of T , denoted by r(T ), as follows: . It had been proved by C. Ehresmann [6] that • v ≤ w with respect to the Bruhat order in the symmetric group S n if and only if T i,j (v) ≤ T i,j (w) for all 1 ≤ i < j ≤ n − 1.
It follows from an improved tableau criterion for Bruhat order on the symmetric group [2] that 15 15 It has been proved in [2], Corollary 5, that the Ehresmann criterion stated above is equivalent to either the criterion T i,j ≤ T (2) i,j f or all j such that w j > w j+1 and 1 ≤ i ≤ j, or that T i,j ≤ T i,j f or all j ∈ {1, 2, . . . , n − 1}\{k | v k > v k+1 } and 1 ≤ i ≤ j.