1 a Basis of the Basic Sl ( 3 , C ) ∼ - Module

Abstract

J. Lepowsky and R. L. Wilson initiated the approach to combinatorial Rogers-Ramanujan type identities via the vertex operator constructions of representations of affine Lie algebras. In this approach the first new combinatorial identities were discovered by S. Capparelli through the construction of the level 3 standard A (2) 2 -modules. We obtained several infinite series of new combinatorial identities through the construction of all standard A (1) 1 -modules; the identities associated to the fundamental modules coincide with the two Capparelli identities. In this paper we extend our construction to the basic A (1) 2 -module and, by using the principal specialization of the Weyl-Kac character formula, we obtain a Rogers-Ramanujan type combinatorial identity for colored partitions. The new combinatorial identity indicates the next level of complexity which one should expect in Lepowsky-Wilson’s approach for affine Lie algebras of higher ranks, say for A (1) n , n ≥ 2, in a way parallel to the next level of complexity seen when passing from the Rogers-Ramanujan identities (for modulus 5) to the Gordon identities for odd moduli ≥ 7. Introduction J. Lepowsky and R. L. Wilson gave in [LW] a Lie-theoretic interpretation and proof of the classical Rogers-Ramanujan identities in terms of representations of the affine Lie algebra g̃ = sl(2,C). The identities are obtained by expressing in two ways the principal characters of vacuum spaces for the principal Heisenberg subalgebra of g̃. The product sides follow from the principally specialized WeylKac character formula; the sum sides follow from the vertex operator construction of bases parametrized by partitions satisfying difference 2 conditions. Very roughly speaking, for a level 3 standard g̃-module L with a highest weight vector v0, Lepowsky and Wilson construct Z-operators {Z(j) | j ∈ Z} which commute with the action of the principal Heisenberg subalgebra, and show that Z(j1)Z(j2) . . . Z(js)v0, j1 ≤ j2 ≤ · · · ≤ js <

Cite this paper

@inproceedings{Primc20011AB, title={1 a Basis of the Basic Sl ( 3 , C ) ∼ - Module}, author={Mirko Primc}, year={2001} }