A bijection which implies Melzer's polynomial identities: the $\chi_{1,1}^{(p,p+1)}$ case

TL;DR

This paper constructs a bijection between lattice paths and partitions to prove Melzer's polynomial identities, which in the limit yield Rogers-Ramanujan-type identities for specific Virasoro characters, confirming longstanding conjectures.

Contribution

It introduces a novel bijection that proves Melzer's polynomial identities and connects them to Rogers-Ramanujan-type identities for Virasoro characters.

Findings

Proof of Melzer's polynomial identities

Reduction to Rogers-Ramanujan-type identities in the limit

Confirmation of conjectures by the Stony Brook group

Abstract

We obtain a bijection between certain lattice paths and partitions. This implies a proof of polynomial identities conjectured by Melzer. In a limit, these identities reduce to Rogers--Ramanujan-type identities for the $\chi_{1,1}^{(p,p+1)}(q)$ Virasoro characters, conjectured by the Stony Brook group.