Partition Identities and Quiver Representations

TL;DR

This paper establishes a novel connection between partition combinatorics and quiver representation theory, providing bijective proofs and new insights into identities related to Dynkin quivers.

Contribution

It introduces a bijective proof of an analogue of Durfee square identity for multipartitions and offers a new proof of Reineke's identity for Dynkin type A quivers.

Findings

Bijective proof of Durfee square identity for multipartitions

New proof of Reineke's identity for Dynkin type A quivers

Connection between partition identities and quiver orbit parametrization

Abstract

We present a particular connection between classical partition combinatorics and the theory of quiver representations. Specifically, we give a bijective proof of an analogue of A. L. Cauchy's Durfee square identity to multipartitions. We then use this result to give a new proof of M. Reineke's identity in the case of quivers $Q$ of Dynkin type $A$ of arbitrary orientation. Our identity is stated in terms of the lacing diagrams of S. Abeasis - A. Del Fra, which parameterize orbits of the representation space of $Q$ for a fixed dimension vector.