Classical partition functions and the U( n+1) Rogers-Selberg identity

Abstract

In this paper we show that after suitable specialization the ‘balanced’ side of the U( n+1) Rogers-Selberg identity gives the generating function for all partitions whose parts differ by at least ( n+1). A similar specialization yields the additional condition that the parts must be ⩾ n+1. The case n=1 is the sum side of the pair of classical Rogers-Ramanujan-Schur identities. Our derivation of this connection between classical partition functions and the U( n+1) Rogers-Selberg identity relies upon partial fraction techniques, Hall-Littlewood polynomials, raising operators, q-Kostka matrices, the Cauchy identity for Schur functions, and generating functions for column-strict plane partitions. One outcome of this work is a new class of symmetric functions H λ ( z; γ, q), analogous to Hall-Littlewood polynomials, that interpo- lates between Schur functions and complete homogeneous symmetric functions. Let ω: Λ→ Λ be the classical involution on the ring of symmetric functions defined by ω( h λ )= e λ . Then the function ω( H λ ( z;∅; q))= E λ ( z;∅; q) leads to explicit q-difference equations, raising operator formulas, and a combinatorial setting for an important nontrivial special case Macdonald's new class of symmetric functions with two free parameters. This connection with Macdonald's work is observed and developed in the paper of A. Garsia.