Partition identities associated with $A_r$-Surface singularities

TL;DR

This paper establishes new partition identities involving three-colored partitions linked to $A_r$ surface singularities, extending classical identities like Rogers-Ramanujan and Andrews-Gordon through arc HP-series.

Contribution

It introduces a family of partition identities connected to $A_r$ singularities, expanding the understanding of their combinatorial and geometric properties.

Findings

Identifies new partition identities involving three colors.

Connects partition identities with $A_r$ surface singularities.

Extends classical Rogers-Ramanujan and Andrews-Gordon identities.

Abstract

We prove a family of partition identities involving integer partitions in three colors. The conditions imposed on the types of partitions appearing in these identities involve constraints that arise in the Rogers-Ramanujan and Andrews-Gordon identities, as well as in their recent extensions. The identities established in this paper are associated with the $A_r$ surface singularities via the arc HP-series, which provides a measure of singularities of algebraic varieties defined using arc spaces.