Hive-type polytopes for quiver multiplicities and the membership problem for quiver moment cones

TL;DR

This paper provides a polyhedral combinatorics approach to quiver representation multiplicities, expressing them as lattice point counts in specific polytopes, and offers an efficient algorithm for the membership problem in quiver moment cones.

Contribution

It introduces a novel polytope construction for quiver multiplicities and applies Tardos' algorithm to solve the membership problem efficiently.

Findings

Quiver multiplicities are represented as lattice points in glued hive polytopes.

The approach enables polynomial-time solutions for the membership problem.

The method leverages known theorems to connect representation theory and polyhedral geometry.

Abstract

Let $Q$ be a bipartite quiver with vertex set $Q_0$ such that the number of arrows between any source vertex and any sink vertex is constant. Let $\beta=(\beta(x))_{x \in Q_0}$ be a dimension vector of $Q$ with positive integer coordinates. Let $rep(Q, \beta)$ be the representation space of $\beta$-dimensional representations of $Q$ and $GL(\beta)$ the base change group acting on $rep(Q, \beta)$ be simultaneous conjugation. Let $K^{\beta}_{\underline{\lambda}}$ be the multiplicity of the irreducible representation of $GL(\beta)$ of highest weight $\underline{\lambda}$ in the ring of polynomial functions on $rep(Q, \beta)$. We show that $K^{\beta}_{\underline{\lambda}}$ can be expressed as the number of lattice points of a polytope obtained by gluing together two Knutson-Tao hive polytopes. Furthermore, this polytopal description together with Derksen-Weyman's Saturation Theorem for quiver semi-invariants allows us to use Tardos' algorithm to solve the membership problem for the moment cone associated to $(Q,\beta)$ in strongly polynomial time.