Half-lattice paths and Virasoro characters

TL;DR

This paper introduces and analyzes half-lattice paths to derive fermionic expressions for Virasoro minimal model characters, providing a uniform treatment of cases and establishing new results for the M(p,2p-1) models.

Contribution

It reformulates half-lattice paths for M(p,2p±1) characters, proves their generating functions match these characters, and derives new fermionic expressions using combinatorial bijections.

Findings

Established weight-preserving bijections between half-lattice paths and RSOS paths.

Proved the generating functions of half-lattice paths are the Virasoro characters.

Derived fermionic expressions for M(p,2p±1) characters, including the first derivation for M(p,2p-1).

Abstract

We first briefly review the role of lattice paths in the derivation of fermionic expressions for the M(p,p') minimal model characters of the Virasoro Lie algebra. We then focus on the recently introduced half-lattice paths for the M(p,2p+/-1) characters, reformulating them in such a way that the two cases may be treated uniformly. That the generating functions of these half-lattice paths are indeed M(p,2p+/-1) characters is proved by describing weight preserving bijections between them and the corresponding RSOS lattice paths. Here, the M(p,2p-1) case is derived for the first time. We then apply the methods of Bressoud and Warnaar to these half-lattice paths to derive fermionic expressions for the Virasoro characters X^{p,2p+/-1}_{1,2} that differ from those obtained from the RSOS paths. This work is an extension of that presented by the third author at the "7th International Conference on Lattice Path Combinatorics and Applications", Siena, Italy, July 2010.