On the Göllnitz-Gordon-Andrews identities via commutative algebra

Abstract

The Göllnitz-Gordon-Andrews identities generalize the classical partition identities discovered independently by H. Göllnitz and B. Gordon. These are Rogers-Ramanujan-type identities involving generating functions of partitions satisfying certain kinds of difference conditions on the one hand and infinite periodic products on the other. In 2021, Afsharijoo provided a commutative algebra proof of the Rogers-Ramanujan-Gordon identities. Building on Afsharijoo's approach, we investigate the Göllnitz-Gordon-Andrews identities using techniques from commutative algebra. More generally, we establish a broader family of identities, of which the Göllnitz-Gordon-Andrews identities arise as special cases. Our approach interprets the associated generating functions in terms of Hilbert-Poincaré series of suitably constructed graded algebras, providing the first commutative algebra framework for these identities.