Combinatorial constructions of modules for infinite-dimensional Lie algebras, I. Principal subspace

Abstract

This is the first of a series of papers studying combinatorial (with no ``subtractions'') bases and characters of standard modules for affine Lie algebras, as well as various subspaces and ``coset spaces'' of these modules. In part I we consider certain standard modules for the affine Lie algebra $\ga,\;\g := sl(n+1,\C),\;n\geq 1,$ at any positive integral level $k$ and construct bases for their principal subspaces (introduced and studied recently by Feigin and Stoyanovsky [FS]). The bases are given in terms of partitions: a color $i,\;1\leq i \leq n,$ and a charge $s,\; 1\leq s \leq k,$ are assigned to each part of a partition, so that the parts of the same color and charge comply with certain difference conditions. The parts represent ``Fourier coefficients'' of vertex operators and can be interpreted as ``quasi-particles'' enjoying (two-particle) statistical interaction related to the Cartan matrix of $\g.$ In the particular case of vacuum modules, the character formula associated with our basis is the one announced in [FS]. New combinatorial characters are proposed for the whole standard vacuum $\ga$-modules at level one.