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

TL;DR

This paper develops combinatorial bases for principal subspaces of standard modules for affine Lie algebras, using partitions with color and charge, leading to new character formulas and insights into quasi-particle structures.

Contribution

It introduces a new combinatorial construction of bases for principal subspaces of affine Lie algebra modules using partitions with color and charge, and derives associated character formulas.

Findings

Constructed bases for principal subspaces using partitions with difference conditions.

Derived new combinatorial character formulas for affine Lie algebra modules.

Interpreted parts as quasi-particles with statistical interactions.

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.