Divisors on complete multigraphs and Donaldson-Thomas invariants of loop quivers

TL;DR

This paper explores the relationship between divisors on complete multigraphs, group actions, and Donaldson-Thomas invariants, revealing new combinatorial and algebraic connections in the context of quiver theory.

Contribution

It provides a novel characterization of divisors on multigraphs and links their orbits to Donaldson-Thomas invariants, also identifying related parking function modules.

Findings

Orbits of divisors are enumerated by Donaldson-Thomas invariants.

Characterization of divisors leads to new combinatorial interpretations.

Identification of parking function modules related to group actions.

Abstract

We study the action of $S_n$ on the set of break divisors on complete multigraphs $K_{n}^m$. We provide an alternative characterization for these divisors, by virtue of which we show that orbits of this action are enumerated by the numerical Donaldson-Thomas invariants of $(m+1)$-loop quivers. Our characterization also allows us to restrict this action to $S_{n-1}$ and we identify the resulting $S_{n-1}$-module as that afforded by $K_n^m$-parking functions.