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 from that obtained in \QCITE{cite}{}{lec3} from the cyclage graph structure on tableaux of type $C_{n}$. We show that such a structure also exists for the tableaux of types $B_{n}$ and $D_{n}$ but can not be simply related to the Kostka-Foulkes polynomials. Finally we give explicit formulas for $K_{λ,μ}(q)$ when $| λ| \leq 3,$ or $n=2$ and $μ=0$.
We investigate the relationship between Kostka-Foulkes polynomials and certain symmetric functions that arise from Garsia and Haglund's study of the q,t-Catalan series.
An iterative formula for the Kostka-Foulkes polynomials is given using the vertex operator realization of the Hall-Littlewood polynomials. The operational formula can handle large Kostka-Foulkes polynomials, and a stability property for the Kostka-Foulkes polynomials is shown. We also use our algorithm to give a formula of $K_{λμ}(t)$ for $μ$ being hook-shaped.
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 …
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.
Generalized Hall-Littlewood polynomials (Macdonald spherical functions) and generalized Kostka-Foulkes polynomials ($q$-weight multiplicities) arise in many places in combinatorics, representation theory, geometry, and mathematical physics. This paper attempts to organize the different definitions of these objects and prove the fundamental combinatorial results from ``scratch'', in a presentation which, hopefully, will be accessible and useful for both the nonexpert and researchers currently working in this very active field. The combinatorics of the affine Hecke algebra plays a central role. …
We establish a Morris type recurrence formula for the root system $C_{n}$.\ Next we introduce cyclage graphs for the corresponding Kashiwara-Nakashima's tableaux and use them to define a charge statistic. Finally we conjecture that this charge may be used to compute the Kostka-Foulkes polynomials for type $C_{n}.$
The Kostka-Foulkes polynomials $K$ related to a root system $ϕ$ can be defined as alternated sums running over the Weyl group associated to $ϕ.$ By restricting these sums over the elements of the symmetric group when $% ϕ$ is of type $B,C$ or $D$, we obtain again a class $\widetilde{K}$ of Kostka-Foulkes polynomials. When $ϕ$ is of type $C$ or $D$ there exists a duality beetween these polynomials and some natural $q$-multiplicities $U$ in tensor product \cite{lec}. In this paper we first establish identities for the $\widetilde{K}$ which implies in particular that they can be decomposed as sum…
We report about results revolving around Kostka-Foulkes and parabolic Kostka polynomials and their connections with Representation Theory and Combinatorics. It appears that the set of all parabolic Kostka polynomials forms a semigroup, which we call {\it Liskova semigroup}. We show that polynomials frequently appearing in Representation Theory and Combinatorics belong to the Liskova semigroup. Among such polynomials we study rectangular $q$-Catalan numbers; generalized exponents polynomials; principal specializations of the internal product of Schur functions; generalized $q$-Gaussian polynomi…
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.
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.
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.
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.
A combinatorial proof of the unimodality of the generalized q-Gaussian coefficients based on the explicit formula for Kostka-Foulkes polynomials is given.
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…
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…
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…
The Gini index is a number that attempts to measure how equitably a resource is distributed throughout a population, and is commonly used in economics as a measurement of inequality of wealth or income. The Gini index is often defined as the area between the "Lorenz curve" of a distribution and the line of equality, normalized to be between zero and one. In this fashion, we will define a Gini index on the set of integer partitions and prove some combinatorial results related to it; culminating in the proof of an identity for the expected value of the Gini index. We will then discuss symmetric …
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.
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.
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…
