Combinatorial constructions of modules for infinite-dimensional Lie algebras, II. Parafermionic space

TL;DR

This paper develops a combinatorial basis for parafermionic spaces associated with affine Lie algebras, generalizing previous approaches and providing new character formulas for standard modules at positive levels.

Contribution

It introduces a new combinatorial basis for parafermionic spaces of affine Lie algebras, extending the Z-algebra approach and connecting to Bethe Ansatz conjectures.

Findings

Constructed a combinatorial basis using colored partitions.

Derived new character formulas for standard modules.

Connected bases to Bethe Ansatz conjectures.

Abstract

The standard modules for an affine Lie algebra $\ga$ have natural subquotients called parafermionic spaces -- the underlying spaces for the so-called parafermionic conformal field theories associated with $\ga.$ We study the case $\ga = \widehat{sl}(n+1,\C)$ for any positive integral level $k \geq 2.$ Generalizing the $\cal Z$-algebra approach of Lepowsky, Wilson and Primc, we construct a combinatorial basis for the parafermionic spaces in terms of colored partitions. The parts of these partitions represent ''Fourier coefficients'' of generalized vertex operators (parafermionic currents) and can be interpreted as statistically interacting quasi-particles of color $i,\;1\leq i \leq n,$ and charge $s,\; 1\leq s \leq k-1.$ From a combinatorial point of view, these bases are essentially identical with the bases for level $k-1$ principal subspaces constructed by the author in [GeI]. In the particular case of the vacuum module, the character (string function) associated with our basis is the formula of Kuniba, Nakanishi and Suzuki [KNS] conjectured in a Bethe Ansatz layout. New combinatorial characters are established for the whole standard vacuum $\ga$-modules.