Cylindrical Combinatorics and Representations of Cherednik Algebras of type A

Abstract

We investigate the representation theory of the rational and trigonometric Cherednik algebra of type $GL_n$ by means of combinatorics on periodic (or cylindrical) skew diagrams. We introduce and study standard tableaux and plane partitions on periodic diagrams, and in particular, compute some generating functions concerning plane partitions, where Kostka polynomials and their level restricted generalization appear. On representation side, we study representations of Cherednik algebras which admit weight decomposition with respect to a certain commutative subalgebra. All the irreducible representations of this class are constructed combinatorially through standard tableaux on periodic diagrams, and this realization as "tableaux representations" provides a new combinatorial approach to the investigation of these representations. As consequences, we describe the decomposition of a tableaux representation as a representation of the degenerate affine Hecke algebra, which is a subalgebra of the Cherednik algebra, and also describe the spectral decomposition of the spherical subspace (the invariant subspace under the action of the Weyl group) of a tableaux representation with respect to the center of the degenerate affine Hecke algebra, In particular, the computation of the character of the spherical subspace is reduced to the computation of the generating function for the set of column strict plane partitions, and we obtain an expression of the characters in terms of Kostka polynomials as announced in math.QA/0508274.