$J$-generalization of the Rogers-Ramanujan-Gordon identities via commutative algebra

TL;DR

This paper extends the commutative algebra proof technique to a broader family of partition identities, including Rogers-Ramanujan-Gordon, by relating generating functions to Hilbert-Poincaré series.

Contribution

It provides a new commutative algebra proof for a generalized family of identities, expanding on Afsharijoo's work and including Rogers-Ramanujan-Gordon as a special case.

Findings

Established a commutative algebra proof for the broader family of identities.

Connected generating functions to Hilbert-Poincaré series of graded algebras.

Generalized the Rogers-Ramanujan-Gordon identities within this framework.

Abstract

The Rogers-Ramanujan-Gordon identities generalize the classical partition identities discovered independently by L. J. Rogers and S. Ramanujan. In 2021, Afsharijoo provided a commutative algebra proof of the Rogers-Ramanujan-Gordon identities. Building on the Afsharijoo's approach, we present a commutative algebra proof of a broader family of identities introduced by Coulson \textit{et al.}, which includes the Rogers-Ramanujan-Gordon identities as a special case. In the proof, we relate the generating functions associated with these identities to the Hilbert-Poincar\'e series of suitably constructed graded algebras.