Kostka-Foulkes polynomials cyclage graphs and charge statistic for the root system $C_{n}$

Nonsymmetric Macdonald polynomials and a refinement of Kostka-Foulkes polynomials

We study the specialization of the type A nonsymmetric Macdonald polynomials at $t=0$ based on the combinatorial formula of Haglund, Haiman, and Loehr. We prove that this specialization expands nonnegatively into the fundamental slide polynomials, introduced by the author and Searles. Using this and weak dual equivalence, we prove combinatorially that this specialization is a positive graded sum of Demazure characters. We use stability results for fundamental slide polynomials to show that this specialization stabilizes and to show that the Demazure character coefficients give a refinement of …

Combinatorics of crystal graphs and Kostka-Foulkes polynomials for the root systems $B_{n},C_{n}$ and $D_{n}.$

We use Kashiwara-Nakashima's combinatorics of crystal graphs associated to the roots sytems $B_{n}$ and $D_{n}$ to extend the results of \QCITE{cite}{}{lec3} and \QCITE{cite}{}{Mor} by showing that Morris type recurrence formulas also exist for the orthogonal root systems. We derive from these formulas a statistic on Kashiwara-Nakashima's tableaux of types $B_{n},C_{n}$ and $D_{n}$ generalizing Lascoux-Sch\UNICODE{0xfc}tzenberger's charge and from which it is possible to compute the Kostka-Foulkes polynomials $K_{λ,μ}(q)$ with restrictive conditions on $(λ,μ)$ . This statistic is different fro…

A positive combinatorial formula for symplectic Kostka-Foulkes polynomials I: Rows

We prove a conjecture of Lecouvey, which proposes a closed, positive combinatorial formula for symplectic Kostka-Foulkes polynomials, in the case of rows of arbitrary weight. To show this, we construct a new algorithm for computing cocyclage in terms of which the conjecture is described. Our algorithm is free of local constraints, which were the main obstacle in Lecouvey's original construction. In particular, we show that our model is governed by the situation in type A. This approach works for arbitrary weight and we expect it to lead to a proof of the conjecture in full generality.

Hall-Littlewood vertex operators and generalized Kostka polynomials

A family of vertex operators that generalizes those given by Jing for the Hall-Littlewood symmetric functions is presented. These operators produce symmetric functions related to the Poincare polynomials referred to as generalized Kostka polynomials in the same way that Jing's operator produces symmetric functions related to Kostka-Foulkes polynomials. These operators are then used to derive commutation relations and new relations involving the generalized Kostka coefficients. Such relations may be interpreted as identities in the (GL(n) x C^*)-equivariant K-theory of the nullcone.

Polynomiality of the q,t-Kostka Revisited

Let $K(q,t)= \|K_{\laμ}(q,t)\|_{\la,μ}$ be the Macdonald q,t-Kostka matrix and $K(t)=K(0,t)$ be the matrix of the Kostka-Foulkes polynomials K_{\laμ}(t). In this paper we present a new proof of the polynomiality of the q,t-Kostka coefficients that is both short and elementary. More precisely, we derive that $K(q,t)$ has entries in \ZZ[q,t] directly from the fact that the matrix $K(t)^{-1}$ has entries in \ZZ[t]. The proof uses only identities that can be found in the original paper [7] of Macdonald.

Spin Kostka 多项式与顶点算子

Spin Kostka polynomials and vertex operators

本文通过顶点算子给出 spin Kostka-Foulkes 多项式的迭代公式，获得平行性质与标记表计数公式，并证实 Aokage 猜想。

Lusztig's q-analogue of weight multiplicity and one-dimensional sums for affine root systems

In this paper we complete the proof of the X=K conjecture, that for every family of nonexceptional affine algebras, the graded multiplicities of tensor products of symmetric power Kirillov-Reshetikhin modules known as one-dimensional sums, have a large rank stable limit X that has a simple expression (called the K-polynomial) as nonnegative integer combination of Kostka-Foulkes polynomials. We consider a subfamily of Lusztig's q-analogues of weight multiplicity which we call stable KL polynomials and denote by KL. We give a type-independent proof that K=KL. This proves that X=KL: the family of…

Generalization of the Gale-Ryser theorem

We prove an inequality for the Kostka-Foulkes polynomials $K_{λ,μ}(q)$. As a corollary, we obtain a nontrivial lower bound for the Kostka numbers and a new proof of the Berenstein-Zelevinsky weight-multiplicity-one-criterium.

Unimodality of generalized Gaussian coefficients

A combinatorial proof of the unimodality of the generalized q-Gaussian coefficients based on the explicit formula for Kostka-Foulkes polynomials is given.

Towards a Combinatorial Model for $q$-weight Multiplicities of Simple Lie Algebras (Extended Abstract)

Kostka-Foulkes polynomials are Lusztig's $q$-analogues of weight multiplicities for irreducible representations of semisimple Lie algebras. It has long been known that these polynomials have non-negative coefficients. A statistic on semistandard Young tableaux with partition content, called \textit{charge}, was used to give a combinatorial formula exhibiting this fact in type $A$. Defining a charge statistic beyond type $A$ has been a long-standing problem. Here, we take a completely new approach based on the definition of Kostka-Foulkes polynomials as an alternating sum over Kostant partition…

Graded characters of modules supported in the closure of a nilpotent conjugacy class

We study the Poincare polynomials of isotypic components of a natural family of graded GL(n)-modules supported in the closure of a nilpotent conjugacy class. These polynomials generalize the Kostka-Foulkes and are q-analogues of Littlewood-Richardson coefficients corresponding to arbitrary tensor products of irreducibles. Many properties and formulas for these polynomials are derived, such as a generalized Morris recurrence, q-Kostant formula, and a conjectural formula in terms of catabolizable tableaux and charge.

Quiver Hall-Littlewood functions and Kostka-Shoji polynomials

For any triple $(i,a,μ)$ consisting of a vertex $i$ in a quiver $Q$, a positive integer $a$, and a dominant $GL_a$-weight $μ$, we define a quiver current $H^{(i,a)}_μ$ acting on the tensor power $Λ^Q$ of symmetric functions over the vertices of $Q$. These provide a quiver generalization of parabolic Garsia-Jing creation operators in the theory of Hall-Littlewood symmetric functions. For a triple $(\mathbf{i},\mathbf{a},μ(\bullet))$ of sequences of such data, we define the quiver Hall-Littlewood function $H^{\mathbf{i},\mathbf{a}}_{μ(\bullet)}$ as the result of acting on $1\inΛ^Q$ by the corres…

Atoms and charge in type $C_2$

We construct atomic decompositions for crystals of type $C_{2}$ and define a charge statistic on them, thus providing positive combinatorial formulas for Kostka-Foulkes polynomials associated to them together with a natural geometric interpretation.

The ABC's of affine Grassmannians and Hall-Littlewood polynomials

We give a new description of the Pieri rule for k-Schur functions using the Bruhat order on the affine type-A Weyl group. In doing so, we prove a new combinatorial formula for representatives of the Schubert classes for the cohomology of affine Grassmannians. We show how new combinatorics involved in our formulas gives the Kostka-Foulkes polynomials and discuss how this can be applied to study the transition matrices between Hall-Littlewood and k-Schur functions.

Atomic decomposition of characters and crystals

Lascoux stated that the type A Kostka-Foulkes polynomials K_{lambda,mu}(t) expand positively in terms of so-called atomic polynomials. For any semisimple Lie algebra, the former polynomial is a t-analogue of the multiplicity of the dominant weight mu in the irreducible representation of highest weight lambda. We formulate the atomic decomposition in arbitrary type, and view it as a strengthening of the monotonicity of K_{lambda,mu}(t). We also define a combinatorial version of the atomic decomposition, as a decomposition of a modified crystal graph. We prove that this stronger version holds in…

Lusztig q-权多重性的多面体不变性

Polytopal invariance for Lusztig's $q$-weight multiplicities

该研究证明了Lusztig q-权多重性对应的凸多面体等距时，其q-多项式相等，还从多面体度量形状构造多项式、分类等距情形，为Kazhdan-Lusztig多项式组合不变性猜想提供了球对称的度量类比。