The method of weighted words revisited

TL;DR

This paper introduces a new variant of the weighted words method to prove complex partition identities, including those from algebraic and combinatorial theories, extending its applicability significantly.

Contribution

It presents a novel variant of the weighted words method and applies it to prove new refinements and generalizations of three advanced partition identities.

Findings

Proved a refinement of Siladić's theorem from vertex operator algebras.

Established a conjectural identity of Primc from crystal base theory.

Unified several generalizations of Schur's theorem through a new identity about coloured overpartitions.

Abstract

Alladi and Gordon introduced the method of weighted words in 1993 to prove a refinement and generalisation of Schur's partition identity. Together with Andrews, they later used it to refine Capparelli's and G\"ollnitz' identities too. In this paper, we present a new variant of this method, which can be used to study more complicated partition identities, and apply it to prove refinements and generalisations of three partition identities. The first one, Siladi\'c's theorem (2002), comes from vertex operator algebras. The second one, a conjectural identity of Primc (1999), comes from crystal base theory. The last one is a very general identity about coloured overpartitions which generalises and unifies several generalisations of Schur's theorem due to Alladi-Gordon, Andrews, Corteel-Lovejoy, Lovejoy and the author.