The widely studied q-polynomial fλ(q), which specializes when q = 1 to fλ, the number of standard Young tableaux of shape λ, has multiple combinatorial interpretations. It represents the dimension of the unipotent representation Sλ q of the finite general linear group GLn(q), it occurs as a special case of the Kostka-Foulkes polynomials, and it gives the generating function for the major index ...

Starting from Jacobi-Trudi’s type determinental expressions for the Schur functions corresponding to types B,C and D, we define a natural q-analogue of the multiplicity [V (λ) : M(μ)] when M(μ) is a tensor product of row or column shaped modules defined by μ. We prove that these q-multiplicities are equal to certain Kostka-Foulkes polynomials related to the root systems C or D. Finally we deriv...

In the ring of symmetric functions the inverse Kostka matrix appears as the transition matrix from the bases given by monomial symmetric functions to the Schur bases. We present both a combinatorial characterization and a recurrent formula for the entries of the inverse Kostka matrix which are different from the results obtained by Egecioglu and Remmel [ER] in 1990. An application to the topolo...

Let W be a complex reflection group, acting on a complex vector space h. Kato has recently introduced the notion of a “Kostka system,” which is a certain collection of finite-dimensional W -equivariant modules for the symmetric algebra on h. In this paper, we show that Kostka systems can be used to construct “exotic” t-structures on the derived category of finite-dimensional modules, and we pro...

We define a bijection from Littlewood–Richardson tableaux to rigged configurations and show that it preserves the appropriate statistics. This proves in particular a quasi-particle expression for the generalized Kostka polynomials KλR(q) labeled by a partition λ and a sequence of rectangles R. The generalized Kostka polynomials are q-analogues of multiplicities of the irreducible GL(n,C)-module...

The aim of this paper is to show how to multithread a compute intensive application in mathematics (Group Theory) for an institutional Desktop Grid platform coordinated by a meta-grid middleware named BonjourGrid which is a fully decentralized Desktop Grid middleware. The paper is twofold: first of all, it shows how to multithread a sequential program for a multicore CPU which participates in t...

We introduce a generalization of the classical Hall–Littlewood and Kostka–Foulkes polynomials to all symmetrizable Kac–Moody algebras. We prove that these Kostka–Foulkes polynomials coincide with the natural generalization of Lusztig’s t-analog of weight multiplicities, thereby extending a theorem of Kato. For g an affine Kac–Moody algebra, we define t-analogs of string functions and use Chered...

In this article, we develop a framework to study the large deviation principle for matrix models and their quantized versions, by tilting measures using limits of spherical integrals obtained in [46,47]. As examples, obtain 1. empirical distribution diagonal entries $UB_NU^*$, sequence $N\times N$ matrices $B_N$ unitary Haar distributed $U$; 2. upper bound eigenvalue $A_N+UB_NU^*$, two sequence...