Affine Hecke algebras and generalized standard Young tableaux

This paper introduces calibrated representations for affine Hecke algebras and classifies and constructs all finite dimensional irreducible calibrated representations. The primary technique is to provide indexing sets for controlling the weight space structure of finite dimensional modules for the affine Hecke algebra. Using these indexing sets we show that (1) irreducible calibrated representations are indexed by skew local regions, (2) the dimension of an irreducible calibrated representation is the number of chambers in the local region, (3) each irreducible calibrated representation is constructed explicitly by formulas which describe the action of the generators of the affine Hecke algebra on a specific basis in the representation space. The indexing sets for weight spaces are generalizations of standard Young tableaux and the construction of the irreducible calibrated affine Hecke algebra modules is a generalization of A. Young's seminormal construction of the irreducible representations of the symmetric group. In this sense Young's construction has been generalized to arbitrary Lie type

Combinatorial Representation Theory

We attempt to survey the field of combinatorial representation theory, describe the main results and main questions and give an update of its current status. We give a personal viewpoint on the field, while remaining aware that there is much important and beautiful work that we have not been able to mention

A probabilistic interpretation of the Macdonald polynomials

The two-parameter Macdonald polynomials are a central object of algebraic combinatorics and representation theory. We give a Markov chain on partitions of k with eigenfunctions the coefficients of the Macdonald polynomials when expanded in the power sum polynomials. The Markov chain has stationary distribution a new two-parameter family of measures on partitions, the inverse of the Macdonald weight (rescaled). The uniform distribution on permutations and the Ewens sampling formula are special cases. The Markov chain is a version of the auxiliary variables algorithm of statistical physics. Properties of the Macdonald polynomials allow a sharp analysis of the running time. In natural cases, a bounded number of steps suffice for arbitrarily large k

Affine Hecke algebras and the Schubert calculus

Using a combinatorial approach which avoids geometry, this paper studies the ring structure of K_T(G/B), the T-equivariant K-theory of the (generalized) flag variety G/B. Here the data is a complex reductive algebraic group (or symmetrizable Kac-Moody group) G, a Borel subgroup B, and a maximal torus T, and K_T(G/B) is the Grothendieck group of T-equivariant coherent sheaves on G/B. We prove "Pieri-Chevalley" formulas for the products of a Schubert class by a homogeneous line bundle (dominant or anti-dominant) and for products of a Schubert class by a codimension 1 Schubert class. All of these Pieri-Chevalley formulas are given in terms of the combinatorics of the Littelmann path model. We give explicit computations of products of Schubert classes for the rank two cases and this data allows us to make a "positivity conjecture" generalizing the theorems of Brion and Graham, which treat the cases K(G/B) and H_T^*(G/B), respectively