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 generalize a construction of Dunkl, obtaining a wide class intertwining functions on the symmetric group and a related family of multidimensional Hahn polynomials. Following a suggestion of Vilenkin and Klymik, we develop a tree-method approach for those intertwining functions. We also give a group theoretic proof of the relation between Hahn polynomials and Clebesh-Gordan coefficients, give...

The spectral decomposition of the path space of the vertex model associated to the level l representation of the quantized affine algebra Uq(ŝln) is studied. The spectrum and its degeneracy are parametrized by skew Young diagrams and what we call nonmovable tableaux on them, respectively. As a result we obtain the characters for the degeneracy of the spectrum in terms of an alternating sum of s...

We provide a new description of the Pieri rule of the homology of the affine Grassmannian and an affine analogue of the charge statistics in terms of bounded partitions. This makes it possible to extend the formulation of the Kostka–Foulkes polynomials in terms of solvable lattice models by Nakayashiki and Yamada to the affine setting. Résumé. Nous proposons une nouvelle description de la règle...

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 w...

and Energy Functions in Solvable Lattice Models Atsushi Nakayashiki and Yasuhiko Yamada Graduate School of Mathematics, Kyushu University Abstract The relation between the charge of Lascoux-Schuzenberger and the energy function in solvable lattice models is clari ed. As an application, A.N.Kirillov's conjecture on the expression of the branching coe cient of c sln=sln as a limit of Kostka polyn...

An iterative formula for the Kostka–Foulkes polynomials is given using vertex operator realization of Hall–Littlewood polynomials. The operational can handle large polynomials, and a stability property shown. We also use our algorithm to give $$K_{\lambda \mu }(t)$$ $$\mu $$ being hook-shaped.