Ghost series and a motivated proof of the Andrews-Bressoud identities

TL;DR

This paper introduces a new method using 'ghost series' and 'shelves' of formal series to provide a motivated proof of the Andrews-Bressoud identities for even moduli, extending previous proofs for related identities.

Contribution

It develops a novel framework with 'ghost series' and 'shelves' that enables a motivated proof of the Andrews-Bressoud identities, advancing the understanding of partition identities.

Findings

Introduces 'ghost series' and 'shelves' for proof construction

Provides a new motivated proof of Andrews-Bressoud identities

Lays groundwork for vertex-algebraic categorification insights

Abstract

We present what we call a "motivated proof" of the Andrews-Bressoud partition identities for even moduli. A "motivated proof" of the Rogers-Ramanujan identities was given by G. E. Andrews and R. J. Baxter, and this proof was generalized to the odd-moduli case of Gordon's identities by J. Lepowsky and M. Zhu. Recently, a "motivated proof" of the somewhat analogous G\"{o}llnitz-Gordon-Andrews identities has been found. In the present work, we introduce "shelves" of formal series incorporating what we call "ghost series," which allow us to pass from one shelf to the next via natural recursions, leading to our motivated proof. We anticipate that these new series will provide insight into the ongoing program of vertex-algebraic categorification of the various "motivated proofs."