Symmetry, Integrability and Geometry: Methods and Applications Vector-Valued Jack Polynomials from Scratch

Vector-valued Jack polynomials associated to the symmetric group ${\mathfrak S}_N$ are polynomials with multiplicities in an irreducible module of ${\mathfrak S}_N$ and which are simultaneous eigenfunctions of the Cherednik-Dunkl operators with some additional properties concerning the leading monomial. These polynomials were introduced by Griffeth in the general setting of the complex reflections groups $G(r,p,N)$ and studied by one of the authors (C. Dunkl) in the specialization $r=p=1$ (i.e. for the symmetric group). By adapting a construction due to Lascoux, we describe an algorithm allowing us to compute explicitly the Jack polynomials following a Yang-Baxter graph. We recover some properties already studied by C. Dunkl and restate them in terms of graphs together with additional new results. In particular, we investigate normalization, symmetrization and antisymmetrization, polynomials with minimal degree, restriction etc. We give also a shifted version of the construction and we discuss vanishing properties of the associated polynomials.


Introduction
The Yang-Baxter graphs are used to study Jack polynomials. In particular, these objects have been investigated in this context by Lascoux [14] (see also [15,16] for general properties about Jack and Macdonald polynomials). Vector-valued Jack polynomials are associated with irreducible representations of the symmetric group S N , that is, to partitions of N . A Yang-Baxter graph is a directed graph with no loops and a unique root, whose edges are labeled by generators of a certain subsemigroup of the extended affine symmetric group. In this paper the vertices are labeled by a pair consisting of a weight in N N and the content vector of a standard tableau. The weights are the labels of monomials which are the leading terms of polynomials, and the tableaux all have the same shape. There is a vector-valued Jack polynomial associated with each vertex. These polynomials are special cases of the polynomials introduced by Griffeth [7,8] for the family of complex reflection groups denoted by G (r, p, N ) (where p|r). This is the group of unitary N × N matrices such that their nonzero entries are r th roots of unity, the product of the nonzero entries is a (r/p) th root of unity, and there is exactly one nonzero entry in each row and each column. The symmetric group is the special case G (1, 1, N ). The vector space in which the Jack polynomials take their values is equipped with the nonnormalized basis described by Young, namely, the simultaneous eigenvectors of the Jucys-Murphy elements.
The labels on edges denote transformations to be applied to the objects at a vertex. Vectorvalued Jack polynomials are uniquely determined by their spectral vector, the vector of eigenvalues under the (pairwise commuting) Cherednik-Dunkl operators. This serves to demonstrate the claim that different paths from one vertex to another produce the same result, a situation which is linked to the braid or Yang-Baxter relations. These refer to the transformations.
Following Lascoux [14] we define the monoid S N , a subsemigroup of the affine symmetric group, with generators {s 1 , s 2 , . . . , s N −1 , Ψ} and relations: The relations s 2 i = 1 do not appear in this list because the graph has no loops. The main objects of our study are polynomials in x = (x 1 , . . . , x N ) ∈ R N with coefficients in Q (α), where α is a transcendental (indeterminate), and with values in the S N -module corresponding to a partition λ of N . The Yang-Baxter graph G λ is a pictorial representation of the algorithms which produce the Jack polynomials starting with constants. The generators of S N correspond to transformations taking a Jack polynomial to an adjacent one. At each vertex there is such a polynomial, and a 4-tuple which identifies it. The 4-tuple consists of a standard tableau denoting a basis element of the S N -module, a weight (multi-index) describing the leading monomial of the polynomial, a spectral vector, and a permutation, essentially the rank function of the weight. The spectral vector and permutation are determined by the first two elements. For technical reasons the standard tableaux are actually reversed, that is, the entries decrease in each row and each column. This convention avoids the use of a reversing permutation, in contrast to Griffeth's paper [8] where the standard tableaux have the usual ordering.
The symmetric and antisymmetric Jack polynomials are constructed in terms of certain subgraphs of G λ . Furthermore the graph technique leads to the definition and construction of shifted inhomogeneous vector-valued Jack polynomials.
Here is an outline of the contents of each section. Section 2 contains the basic definitions and construction of the graph G λ . The presentation is in terms of the 4-tuples mentioned above. It is important to note that not every possible label need appear on edges pointing away from a given vertex: if the e weight at the vertex is v ∈ N N then the transposition (i, i + 1) (labeled by s i ) can be applied only when v [i] ≤ v [i + 1], that is, when the resulting weight is greater than or equal to v in the dominance order. The action of the affine element Ψ is given by vΨ = (v [2] , v [3] , . . . , v [N ] , v [1] + 1).
The Murphy basis for the irreducible representation of S N along with the definition of the action of the simple reflections (i, i + 1) on the basis is presented in Section 3 Also the vectorvalued polynomials, their partial ordering, and the Cherednik-Dunkl operators are introduced here.
Section 4 is the detailed development of Jack polynomials. Each edge of the graph G λ determines a transformation that takes the Jack polynomial associated with the beginning vertex to the one at the ending vertex of the edge. There is a canonical pairing defined for the vectorvalued polynomials; the pairing is nonsingular for generic α and the Cherednik-Dunkl operators are self-adjoint. The Jack polynomials are pairwise orthogonal for this pairing and the squared norm of each polynomial can be found by use of the graph.
In Section 5 we investigate the symmetric and antisymmetric vector-valued Jack polynomials in relation with connectivity of the Yang-Baxter graph whose affine edges have been removed. Also, in this section one finds the method of producing coefficients so that the corresponding sum of Jack polynomials is symmetric or antisymmetric. The idea is explained in terms of certain subgraphs of G λ .
Vertices of G λ satisfying certain conditions may be mapped to vertices of a graph related to S M , M < N , by a restriction map. This topic is the subject of Section 6. This section also describes the restriction map on the Jack polynomials.
In Section 7 the shifted vector-valued Jack polynomials are presented. These are inhomogeneous and the parts of highest degree coincide with the homogeneous Jack polynomials of the previous section. The construction again uses the Yang-Baxter graph G λ ; it is only necessary to change the operations associated with the edges.
Throughout the paper there are numerous figures to concretely illustrate the structure of the graphs.
2 Yang-Baxter type graph associated to a partition 2

.1 Sorting a vector
Consider a vector v ∈ N N , we want to compute the unique decreasing partition v + , which is in the orbit of v for the action of the symmetric group S N acting on the right on the position, using the minimal number of elementary transpositions s i = (i i + 1).
If v is a vector we will denote by v[i] its ith component. Each σ ∈ S N will be associated to the vector of its images [σ(1), . . . , σ(N )]. Let σ be a permutation, we will denote ℓ(σ) = min{k : σ = s i 1 · · · s i k } the length of the permutation. By a straightforward induction one finds: The permutation σ v is obtained by a standardization process: we label with integer from 1 to N the positions in v from the largest entries to the smallest one and from left to right. The definition of σ v is compatible with the action of S N in the following sense: , we obtain the vector of contents by labeling the numbers of the diagonals We construct a Yang-Baxter-type graph with vertices labeled by 4-tuples (τ, ζ, v, σ), where τ is a RST, ζ is a vector of length N with entries in Z[α] (ζ will be called the spectral vector), v ∈ N N and σ ∈ S N , as follows: First, consider a RST of shape λ and write a vertex labeled by the 4-tuple (τ, CT τ , 0 N , [1, . . . , N ]). Now, we consider the action of the elementary transposition of S N on the 4-tuple given by where τ (i,j) denotes the filling obtained by permuting the values i and j in τ . Consider also the affine action given by Definition 2.9. If λ is a partition, denote by τ λ the tableau obtained by filling the shape λ from bottom to top and left to right by the integers {1, . . . , N } in the decreasing order.
The graph G λ is an infinite directed graph constructed from the 4-tuple called the root and adding vertices and edges following the rules 1. We add an arrow labeled by s i from the vertex (τ, and τ is obtained from τ ′ by interchanging the position of two integers k < ℓ such that k is at the south-east of ℓ (i.e. CT τ (k) ≥ CT τ (ℓ) + 2).
2. We add an arrow labeled by Ψ from the vertex (τ,

We add an arrow
An arrow of the form will be called a step. The other arrows will be called jumps, and in particular an arrow (τ, ζ, v, σ) ∅ s i will be called a fall; the other jumps will be called correct jumps.
As usual a path is a sequence of consecutive arrows in G λ starting from the root and is denoted by the sequence if the labels of its arrows. Two paths P 1 = (a 1 , . . . , a k ) and P 2 = (b 1 , . . . , b ℓ ) are said to be equivalent (denoted by P 1 ≡ P 2 ) if they lead to the same vertex.
We remark that from Proposition 2.3, in the case v[i] = v[i + 1], the part 1 of Definition 2.9 is equivalent to the following statement: τ ′ is obtained from τ by interchanging are not allowed.
For a reverse standard tableau τ of shape λ, a partition of N , let where rw(i, τ ) is the row of τ containing i (also we denote the column containing i by cl(i, τ )). Then a correct jump from τ to τ ′ implies inv(τ ′ ) = inv(τ )+1 (the entries σ[i] and σ[i+1] = σ[i]+1 are interchanged in τ to produce τ ′ ). Thus the number of correct jumps in a path from the root to (τ, ζ, v, σ) equals inv(τ ) − inv(τ λ ). So we consider the number of steps in a path from 0 N to v; 1 recall that one step There is a symmetry relation: The above formula can be written as It remains to show that S(vΨ) = S(v) (because |vΨ| = |v| + 1). Note (vΨ)[N ] = v[1] + 1. Then This completes the proof.
As a straightforward consequences, Proposition 2.13 implies Corollary 2.14. All the paths joining two given vertices in G λ have the same length.
This suggests that some properties could be shown by induction on the common length of all the all the paths joining two given vertices.
For a given 4-tuple (τ, ζ, v, σ) the values of ζ and σ are determined by those of τ and v, as shown by the following proposition.
Proof . We prove the result by induction on the length k of a path (a 1 , . . . , a k ) (from Corollary 2.14 all the paths have the same length) from the root to (τ, ζ, v, σ) and set Suppose that a k = Ψ is the affine operation. More precisely, τ = τ ′ , ζ = ζ ′ Ψ α , v = v ′ Ψ and σ = σ ′ [2, . . . , N, 1]. Using the induction hypothesis one has σ ′ = σ v ′ and ζ ′ Hence, Proposition 2.4 gives σ = σ v ′ Ψ = σ v . Suppose that i < N then If i = N then again Suppose now that a k is not an affine operation. Using the induction hypothesis one has , and again the result is straightforward. And similarly when j = i + 1 one finds the correct value for ζ[i + 1].
As a consequence, Corollary 2.17. Let (τ, v) be a pair constituted with a RST τ of shape λ (a partition of N ) and a vector v ∈ N N . Then there exists a unique vertex in G λ labeled by a 4-tuple of the form (τ, ζ, v, σ). We will denote V τ,ζ,v,σ := (τ, v).
We point out that all the information can be retrieved from the spectral vector ζ -the coefficients of α give v, the rank function of v gives σ, and the constants in the spectral vector give the content vector which does uniquely determine the RST τ .
Definition 2.18. We define the subgraph G τ as the graph obtained from G λ by erasing all the vertices labeled by RST other than τ and the associated arrows. Such a graph is connected.
Note that the graph G λ is the union of the graphs G τ connected by jumps. Furthermore, if G τ and G τ ′ are connected by a succession of jumps then there is no step from G τ ′ to G τ . Example 2.19. In Fig. 1, the graph G 21 is constituted with the two graphs G 1 32 and G 2 31 connected by jumps (in blue).
3 Vector-valued polynomials 3.1 About the Young seminormal representation of the symmetric group We consider the space V λ spanned by reverse tableaux of shape λ and the (Young) action of the symmetric group as defined by Murphy 2 in [17] by is always a reverse standard tableau when τ is a reverse standard tableau.
Murphy showed [17] that the RST are the simultaneous eigenfunctions of the Jucys-Murphy elements: where s ij denotes the transposition exchanging i and j. More precisely: As usual, a polynomial representation for the Murphy action on the RST can be computed through the Yang-Baxter graph. We start from τ λ and we construct the associated polynomial in the variables t 1 , . . . , t N : where τ [i, j] denotes the integer belonging at the column i and the row j in τ . Such a polynomial is a simultaneous eigenfunction of the Jucys-Murphy idempotents: Suppose that P τ is the polynomial associated to τ . Suppose also that 0 < b τ [i] < 1. Hence, the polynomial P τ (i,i+1) is obtained from the polynomial P τ by acting with s i − b τ [i] (with the standard action of the transposition s i on the variables t j ). [13], Lascoux simplified the Young construction by having recourse to the covariant algebra (of S N ) C[x 1 , . . . , x N ]/Sym + where Sym + is the ideal generated by symmetric functions without constant terms. Note that the covariant algebra is isomorphic to the regular representation C[S N ]. In the aim to adapt his construction to our notations, we replace each polynomial with its dominant monomial represented by the vectors of its exponents. The vector associated to the root of the graph is the vector exponent of the leading monomial in the product of the Vandermonde determinants associated to each column and is obtained by putting the number of the row minus 1 in the corresponding entry. In fact, the covariant algebra being isomorphic to the regular representation of S N , the computation of the polynomials is completely encoded by the action of the symmetric group on the leading monomials, as shown in the following example. Observe that we do not replace the representation by the orbit of the leading monomial (since the space generated by the orbit is in general bigger), but we consider the projection which completely determines the elements. [21010] [12100] [21100] For instance, one has From the construction, the leading monomial of P τ is the product of all the t rw(i,τ )−1 i . For example, the leading monomial in P 51 732 9864

Definition and dominance properties of vector-valued polynomials
Consider the space where Tab(N ) denotes the set of the reverse standard tableaux on {1, . . . , N }.
The algebra C[S N ] ⊗ C[S N ] acts on these spaces by commuting the vector of the powers on the variables on the left component and the action on the tableaux defined by Murphy (equation (3.1)) on the right component.
For simplicity we will denote By abuse of language x v,τ will be referred to as a polynomial. Note that the space M λ is spanned by the set of polynomials which can be naturally endowed with the order ¡ defined by Example 3.6.

031, 2
The partial order ¢ will provide us a relevant dominance notion.
Definition 3.7. The monomial x v,τ is the leading monomial of a polynomial P if and only if P can be written as As in [9], we define Ψ := (θ ⊗ θ)x N , with θ = s 1 s 2 · · · s N −1 . The following proposition describes the transformation properties of leading monomials with respect to the s i and Ψ.
Proposition 3.8. Suppose that x v,τ is the leading monomial in P then 2.
x vΨ,τ is the leading monomial in P Ψ.

Dunkl and Cherednik-Dunkl operators for vector-valued polynomials
We define the Dunkl operators where s ij denotes the transposition which exchanges i and j and This definition is the same as in [4], but our operators act on their left. One has Lemma 3.9. If D i denotes the Dunkl operator, one has Proof . Straightforward from the definition of D i and the equalities

The Cherednik-Dunkl operators are pairwise commuting operators defined by [4]
We do not repeat the proof of the commutation [U i , U j ] = 0 which can be found in [4]. But, as we will see in the next section, this property is not used to prove the existence of the vector-valued Jack polynomials. One has Lemma 3.10.
The three identities are of the same type. We prove only the first one which follows from the equalities The affine operator Ψ has the following commutation properties with the Dunkl operators: As a consequence, one finds.
The action on the RST is given by Proof . One has where ω i := For convenience, defineξ i := αU i − α. From the preceding lemmas, one obtains Proposition 3.14.

Nonsymmetric vector-valued Jack polynomials
In this section we recover the construction, due to one of the authors [4], of a basis of vectorvalued polynomials J v,τ . This construction belongs to a large family of vector-valued Jack polynomials associated to the complex reflection groups G(r, 1, n) defined by Griffeth [8]. We will denote by ζ v,τ their associated spectral vectors. We will see also that many properties of this basis can be deduced from the Yang-Baxter structure.

Yang-Baxter construction associated to G λ
Let λ be a partition and G λ be the associated graph. We construct the set of polynomials (J P ) P path in G λ using the following recursive rules: where the vector ζ is defined by One has the following theorem.
Theorem 4.1. Let P = [a 0 , . . . , a k ] be a path in G λ from the root to (τ, ζ, v, σ). The polynomial J P is a simultaneous eigenfunctions of the operatorsξ i whose leading monomial is x v,τ . Furthermore, the eigenvalues ofξ i associated to J P are equal to ζ[i]. Consequently J P does not depend on the path, but only on the end point (τ, ζ, v, σ), and will be denoted by J v,τ . The family (J v,τ ) v,τ forms a basis of M λ of simultaneous eigenfunctions of the Cherednik operators.
Furthermore, if P leads to ∅ then J P = 0.
Proof . We will prove the result by induction on the length k. If k = 0 then the result follows from Proposition 3.13. Suppose now that k > 0 and let By induction, J [a 1 ,...,a k−1 ] is a simultaneous eigenfunctions of the operatorsξ i such that the associated vector of eigenvalues is given by and the leading monomial is The leading monomial is a consequence of Proposition 3.8.
Suppose now that a k = s i is a non affine arrow, then ζ = ζ ′ s i , v = v ′ s i and .
Let us examine the leading monomials. First, suppose that a k = s i is a step then τ = τ ′ and σ v = σ v ′ s i . From Proposition 3.8, the leading monomial in J P equals the leading term in And the leading monomial is Q = x v,τ as expected. This proves the first part of the theorem and that the family (J v,τ ) v,τ forms a basis of M λ of simultaneous eigenfunctions of the Cherednik operators. Finally . But clearly, the two polynomials are eigenfunction of the Cherednik operators with different eigenvalues from the cases j = i and j = i + 1. This proves that J P = 0.
As a consequence, we will consider the family of polynomials (J v,τ ) v,τ indexed by pairs (v, τ ) where v ∈ N N is a weight and τ is a tableau. The recursive rules of this section first appeared in [8]. The Lemma 5.3 and the Yang-Baxter graph constitute essentially what Griffeth called calibration graph in that paper.

Partial Yang-Baxter-type construction associated to G τ
To compute an expression for a polynomial J v,τ it suffices to find the good path in the subgraph G τ as shown by the following examples. Example 4.4. For the trivial representation (i.e., λ has a single part), note that the Cherednik operators (in [14]) have the same eigenspaces as the Cherednik-Dunkl operators U i (in [4]). In the notations of [14], ξ i reads where ∂ ij denotes the divided difference on the variables x i and x j . Noting that      (when reading s i ).
In conclusion, the computation of vector-valued Jack for a given RST is completely independent of the computations of the vector-valued Jack indexed by the other RST with the same shape.

Normalization
The space V λ spanned by the RST τ of the same shape λ is naturally endowed (up to a multiplicative constant) by S N -invariant scalar product , 0 with respect to which the RST are pairwise orthogonal. As in [4], we set As in [4], we consider the contravariant form , on the space M λ which is the symmetric S N -invariant form extending , 0 and such that the Dunkl operator D i is the adjoint to the multiplication by x i (see appendix in [4] for more details).
The operator x i D i is self adjoint and the adjoint of σ ∈ S N is σ −1 . Since s ij = s −1 ij is self adjoint, U i is self-adjoint for the form , and the polynomials J v,τ are pairwise orthogonal.
Let us compute their squared norms ||J v,τ || 2 (the bilinear form is nonsingular for generic α and positive definite for α in some subset of R [6]). The method is essentially the same as in [4] and we show that the result can be read in the Yang-Baxter graph. More precisely, one has Proposition 4.7.
For instance:

Symmetrization and antisymmetrization
In [2], Baker and Forrester investigated the coefficients and the norm of the symmetric Jack polynomials by symmetrizing the nonsymmetric Jack polynomials. The symmetrization method was used in [5] for the polynomials associated with the complex groups G(r, p, N ). In this section, we generalize their results and obtain symmetric and antisymmetric vector-valued Jack polynomials.

Non-affine connectivity
Let us denote by H λ the graph obtained from G λ by removing the affine edges, all the falls and the vertex ∅. The purpose of this section is to investigate the connected components of H λ .
Recall that v + is the unique decreasing partition obtained by permuting the entries of v.
This shows that the connected components of H λ are indexed by the T (τ, µ) where µ is a partition. Definition 5.3. We will denote by H T the connected component associated to T in H λ . The component H T will be said to be 1-compatible if T is a column-strict tableau. The component H T will be said to be (−1)-compatible if T is a row-strict tableau.    We we use the following result in the sequel, its proof is easy and left to the reader. The following definition is used to find a RST corresponding to a filling of a shape.
Definition 5.6. Let T be a filling of shape λ, the standardization std(T ) of T is the reverse standard tableau with shape λ obtained by the following process: We will denote by λ T the unique partition obtained by sorting in the decreasing order all the entries of T . Note that each H T has a unique sink (that is a vertex with no outward edge) and this vertex is labeled by (std(T ), ζ T , λ T , Id ) for a certain vector ζ T and a unique root.
The sink is denoted by a red disk and the root by a green disk.

Symmetric and antisymmetric Jack polynomials
For convenience, let us define: Denote also, J ∅ := 0. and Let H T be a 1-compatible component of G λ . For each vertex (τ, ζ, v, σ) of H T , we define the coefficient E v,τ by the following induction: 1. E v,τ = 1 if there is no arrows of the form The symmetric group acts on the spectral vectors ζ by permuting their components. Hence the value of E v,τ does not depend on the path used for its computation and the E v,τ are well defined. Indeed, it suffices to check that the definition is compatible with the commutations s i s j = s j s i with |i − j| > 1 and the braid relations s i s i+1 s i = s i+1 s i s i+1 .
Let us first prove the compatibility with the commutation relations. Suppose with |i − j| > 1 and But, since the symmetric group acts on ζ by permuting its components, one has Hence, , and the definition of E v,τ is compatible with the commutations. Now, let us show that the definition is compatible with the braid relations and set Since the symmetric group acts on ζ by permuting its components, one has Hence, , and the definition is compatible with the braid relations. Define the symmetrization operator We will say that a polynomial is symmetric if it is invariant under the action of s i ⊗ s i for each i < N . 3. More precisely, when H T is 1-compatible, the polynomial Proof . 1. Let us prove the first assertion by induction on the length of a path from (τ, ζ, v, σ) where − means that the sum is over the pairs (τ, v) such that there exists an arrow where 0 means that the sum is over the pairs (τ, v) such that there exists an arrow Since, H T is 1-compatible Proposition 5.5 implies that i and i + 1 are in the same row. Hence, Hence, equalities (5.1) and (5.2) imply This proves that ( J 1100, 21 43 is symmetric. Let H T be a connected component, denote by root(T ) the only vertex of H T without inward edge and by sink(T ) = (std(T ), ζ T , λ T , Id) the only vertex of H T without outward edge. Denote by #H T the number of vertices of H T . The following proposition allows to compare the polynomial J T to the symmetrization of J root(T ) .
Proposition 5.11. One has Proof . It suffices to compare the coefficient of J sink(T ) in J T and in J root(T ) .S. The coefficient #H is the order of the stabilizer of λ T . The leading monomial of J sink(T ) does not appear in any other J v,τ so its coefficient in the symmetrization of J root(T ) equals the order of the stabilizer.
Let H T be a (−1)-compatible component of G λ . For each vertex (τ, ζ, v, σ) of H T , we define the coefficient F v,τ by the following induction: Again the F v,τ are well defined since the symmetric group acts on the spectral vectors by permuting their components. Define also the antisymmetrization operator We will say that a polynomial is antisymmetric if it vanishes under the action of 1 − s i ⊗ s i for each i < N .

1.
Let H T be a connected component of G λ . For each vertex (τ, ζ, v, σ) of H T , the polynomial J v,τ A equals J λ T ,std(T ) A up to a multiplicative constant.
3. More precisely, when H T is (−1)-compatible, the polynomial The polynomial is antisymmetric.
And, as in the symmetric case, one has: Proposition 5.14. One has

Normalization
As a consequence of Proposition 4.7, one deduces the following result using Theorems 5.9 and 5.12.
Corollary 5.15. Let H T be a connected component and (τ, ζ, v, σ) be a vertex of H T . Denote by ℓ T τ,v the length of a path from root(T ) to (τ, ζ, v, σ). One has, From Theorems 5.9 and 5.12, vector-valued symmetric and antisymmetric Jack polynomials are also pairwise orthogonal.

1.
Let H T 1 and H T 2 be two 1-compatible connected components. If T 1 = T 2 then J T 1 , J T 2 = 0.
2. Let H T 1 and H T 2 be two (−1)-compatible connected components. If T 1 = T 2 then Proof . It suffices to remark that from Theorem 5.9 (resp. Theorem 5.12) each J T (resp. J ′ T ) is a linear combination of J v,τ for (τ, ζ, v, σ) vertex in the connected component H T .
In the special cases when H T is ±1-compatible, the value of ||J T || 2 admits a remarkable equality.
Proposition 5.17. One has: 1. If H T is a 1-compatible connected component then Proof . The two cases being very similar, let us only prove the symmetric case. From Proposition 5.11, one has: From Corollary 5.15 and Theorem 5.17, one obtains the surprising equalities: Corollary 5.18. If H T is 1-compatible, one has: If H T is (−1)-compatible, one has: Example 5.19. Consider the graph H 11 00 , the sum (5.3) gives as expected.

Symmetric and antisymmetric polynomials with minimal degree
Since the irreducible characters of S N are real it follows that the tensor product of an irreducible module with itself contains the trivial representation exactly once. The tensor product of the module corresponding to a partition λ with the module for t λ (the transpose) contains the sign representation exactly once. We demonstrate these facts explicitly. Using the concepts from Section 4.1 let be symmetric with (rational) coefficients a (τ ) to be determined. We impose the conditions ζ 1 (s i ⊗ s i ) = ζ 1 for i = 1, . . . , N − 1. Fix some i and split the sum as suggested by equation (3.1) ; thus a (τ ) = c/ τ 2 for some constant c.
Consider the module Vt λ . The transpose map takes each RST τ with shape λ to the RST t τ of shape t λ. Thus bt τ 2 for some τ and i, then − 1 2 ≤ bt τ [i] < 0 and the following transformation rules apply: be antisymmetric with (rational) coefficients a (τ ) to be determined. We impose the conditions ζ det (s i ⊗ s i ) = −ζ det for i = 1, . . . , N − 1. Fix some i and write In the first sum . We can now write down the symmetric and antisymmetric Jack polynomials of lowest degree, by replacing the first factors in ζ 1 and ζ det by the corresponding polynomials P τ (x) and Pt τ (x) (as constructed in Section 3). Let l = ℓ (λ) = t λ [1].
In . In the monomial x v occurs only when τ = std (T 1 ), with coefficient c/ std (T 1 ) 2 . This polynomial is a multiple of J T 1 (see Theorem 5.9).
For the antisymmetric case let = τ [4,3,2] . Let The monomial x v occurs only in the term τ = τ λ (see Definition 2.9). This polynomial is a constant multiple of J ′ T det (see Theorem 5.12). We summarize the results of this section in the following theorem.
Theorem 5.22. The subspace of M λ of the symmetric (resp. antisymmetric) polynomials with minimal degree is spanned by only one generator: the symmetric (resp. antisymmetric) Jack polynomial J T 1 (resp. J T det ).
As a consequence one observes a remarkable property.
Corollary 5.23. The Jack polynomial J T 1 (resp. J T det ) is equal to a polynomial which does not depend on the parameter α multiplied by the global multiplicative constant E sink(T 1 ) (resp. F sink(T det ) ).
Proof . The first part of the sentence is a consequence of Theorem 5.22 since the dimension of the space is 1. The values of the multiplicative constants follow from Theorems 5.9 and 5.12 together with the fact that the coefficient of the leading terms in a Jack polynomials J v,τ is 1 (see Theorem 4.1).
Note also that T 1 (resp. T det ) is not the only tableau for which the corresponding symmetric (resp. antisymmetric) Jack does not depend on α (up to a global multiplicative constant).   Figure 6. The first vertices of the graph G 21 with edges Ψ ′ for M = 2.
Denote by G τ the subgraph of G λ whose root is τ . In particular, one has     with the edges Ψ ′ added.

Restrictions on tableaux
In the sequel, as in [15], we will denote a skew partition by λ/µ.

Knop and Sahi operators for vector-valued polynomials
Let us define the following operators which are the vector-valued versions of the operators defined in [9]: Denote by Φ the operator sending each x i to x i−1 for i > 1 and x 1 to x N − α and T := Φ ⊗ (s 1 s 2 · · · s N −1 ).
Proposition 7.1. The operators ς i satisfy the braid relations and the relations between the ς i and the multiplication by the indeterminates are given by the Leibniz rules: Proof . The Leibniz rules are straightforward from the definition while the braid relations are a direct consequence of 1) the braid relations on the s i and the braid relations on the ∂ i , Since the ς i verify the braid relations, they realize the braid group: given a permutation ω ∈ S N and a reduced decomposition ω = s i 1 · · · s i k , the product ς i 1 · · · ς i k is independent of the choice of the reduced decomposition. We will denote ς ω := ς i 1 · · · ς i k .
Furthermore, the algebra generated by the ς i and the x j is isomorphic to the degenerate Hecke affine algebra generated by the operators s i + ∂ i and the variables.
Our goal is to find a basis of simultaneous eigenvectors of the following operatorŝ These operators commute and play the role of Cherednik elements for our representation of the degenerate Hecke affine algebra. As a consequence, one has the following relations: Furthermore, the RST are simultaneous eigenfunctions of the operatorsξ i . More precisely Proposition 7.3.
Proof . The action ofξ i on polynomials with degree 0 in the x i equals the action of the operatorsξ i . Hence, the result follows from the non-shifted version of the equality (Proposition 3.13).
Straightforwardly, the operators ς i andΨ are compatible with the leading monomials in the following sense: Proposition 7.4. Suppose that P a polynomial such that its highest degree component has the leading monomial x v,τ then then the highest degree component of P ς i has the leading monomial x vs i ,τ .

2.
The highest degree component of PΨ has the leading monomial x vΨ,τ .
3. P = [a 1 , . . . , a k−1 , Ψ] then then As expected one obtains Theorem 7.5. Let P = [a 0 , . . . , a k ] be a path in G λ from the root to (τ, ζ, v, σ). The polyno-mialĴ P is a simultaneous eigenfunction of the operatorsξ i whose leading monomial in the highest degree component is x v,τ . Furthermore, the eigenvalue ofξ i associated toĴ P equals ζ[i].
ConsequentlyĴ P does not depend on the path, but only on the end point (τ, ζ, v, σ), and will be denoted byĴ v,τ . The family (Ĵ v,τ ) v,τ forms a basis of M λ of simultaneous eigenfunctions of the Cherednik operators.
Proof . The proof goes as in Theorem 4.1 using respectively Propositions 7.2, 7.3 and 7.4 instead of Propositions 3.14, 3.13 and 3.8.
In consequence, we will consider the family of polynomials (Ĵ v,τ ) v,τ indexed by pairs (v, τ ) where v ∈ N N is a weight and τ is a tableau.  (see Fig. 9).
Proposition 7.14. Denote by V v,τ the vector whose i-th component is . The result is, now, a direct consequence of Lemma 7.11. The vanishing properties described in Proposition 7.14 are obtained by combining the actions of the s i andΦ on the initial vectors V 0 N ,τ .
Example 7.17. Consider the propagation of vanishing properties described in Fig. 10. Lemma 7.13 and Proposition 7.14 suggest that one can compute other vanishing properties by combining the actions of the s i and Φ + . A general closed formula remained to be found and, unfortunately, the vanishing properties obtained by propagation from V 0 N ,τ are not sufficient to characterize the shifted Jack polynomials.

Conclusion
In this paper we used the Yang-Baxter graph technique to produce a structure describing the nonsymmetric Jack polynomials whose values lie in an irreducible S N -module. The graph is directed with no loops and has exactly one root or base point. Any path joining the root to a vertex is essentially an algorithm for constructing the Jack polynomial at that vertex, and the edges making up the path are the steps of the algorithm. The edges are labeled by the generators of the braid group or by an affine operation.
These techniques are used to analyze restriction to a subgroup S M and also to construct symmetric and antisymmetric Jack polynomials. These are associated with certain subgraphs.
Finally the graph technique is used to construct shifted, or inhomogeneous, vector-valued Jack polynomials.
The theory is independent of the numerical value of the parameter α provided that the eigenspaces of ξ i all have multiplicity one, that is, that no two vertices of the graph have the same spectral vector. Future work is needed to analyze situations where this condition is violated, in particular when α has a singular value, n m : 2 ≤ n ≤ N, m ∈ Z, m n / ∈ Z . There may not be a basis of Jack polynomials for the space of all polynomials. For particular choices of λ there may exist symmetric Jack polynomials of highest weight, that is, those annihilated by It seems plausible that any graph describing such a special case would be significantly different from G λ . As a final remark, note that in the case of the trivial representation, some families of highest weight Jack polynomials have been found (see e.g. [3,10,1]) and related to the theory of the fractional quantum Hall effect [12].