The Rogers-Ramanujan identities, the finite general linear groups, and the Hall-Littlewood polynomials

Abstract

We connect Gordon's generalization of the Rogers-Ramanujan identities with the Hall-Littlewood polynomials and with generating functions which arise in a probabilistic setting in the finite general linear groups. This yields a Rogers-Ramanujan type product formula for the n -+ oo probability that an element of GL(n, q) or Mat(n, q) is semisimple. 1. BACKGROUND AND NOTATION The Rogers-Ramanujan identities are among the most remarkable partition identities in number theory and combinatorics. This paper will be concerned with the following generalization of the Rogers-Ramanujan identities, due to Gordon. Let (x)n denote (1 x)(1 x2)... (1 xn). Theorem 1 ([A, page 111]). For 1 2, and complex x with Ixl O )*'' r0,?i(mod 2k+l) where Nj = nj + + nk-1. Gordon's generalization of the Rogers-Ramanujan identities has been widely studied and appears in many places in mathematics and physics. Andrews [A] discusses combinatorial aspects of these identities. In an important series of papers, Lepowsky and Wilson [LW1], [LW2], [LW3] connect the Gordon identities with affine Lie algebras and structures that they called Z-algebras (later interpreted as parafermion algebras in conformal field theory). Meurman and Primc [MP] solve a problem left open in [LW3], proving the independence of a Z-algebra basis and obtaining a Z-algebra proof of the Gordon identities. Feigin and Frenkel [FF] interpret the Gordon identities as a character formula for the Virasoro algebra. Andrews, Baxter, and Forrester [ABF] and Warnaar [W] relate the Gordon identities with statistical mechanics. For some number theoretic connections see the conference proceedings [AABRR]. We use the following standard notation from the theory of partitions. Call A = (A1,A2,--.) a partition of n = AIX if A1 > A2 > > 0 where the Ai are Received by the editors March 6, 1998. 1991 Mathematics Subject Classification. Primary 20P05, 05E05. ()1999 American Mathematical Society 17 This content downloaded from 207.46.13.142 on Sat, 28 May 2016 06:40:23 UTC All use subject to http://about.jstor.org/terms