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

Calin Chindris, Brett Collins, Daniel Kline

TL;DR

The paper develops a polytopal description for quiver multiplicities $K^{\beta}_{\underline{\lambda}}$ in $n$-complete bipartite quivers by gluing Knutson–Tao hive polytopes, yielding a polytope $\mathcal{P}_{\underline{\lambda}}$ whose lattice points equal $K^{\beta}_{\underline{\lambda}}$. Using quiver exceptional sequences and Derksen–Weyman embedding, the authors express $K^{\beta}_{\underline{\lambda}}$ as a sum of products of Littlewood–Richardson coefficients, culminating in an explicit combinatorial linear program whose feasibility governs positivity. This polytopal model, together with the Saturation Theorem, allows the membership problem for the quiver moment cone $\Delta(Q,\beta)$ to be solved in strongly polynomial time via Tardos’ algorithm, and it provides a constructive pathway to generalizations via Vergne–Walter. The work connects representation-theoretic multiplicities, invariant theory, and polyhedral geometry, yielding both concrete computation tools and theoretical insights into quiver semi-invariants and their stability criteria.

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 $β=(β(x))_{x \in Q_0}$ be a dimension vector of $Q$ with positive integer coordinates. Let $rep(Q, β)$ be the representation space of $β$-dimensional representations of $Q$ and $GL(β)$ the base change group acting on $rep(Q, β)$ be simultaneous conjugation. Let $K^β_{\underlineλ}$ be the multiplicity of the irreducible representation of $GL(β)$ of highest weight $\underlineλ$ in the ring of polynomial functions on $rep(Q, β)$. We show that $K^β_{\underlineλ}$ 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 $(Q,β)$ in strongly polynomial time.

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

TL;DR

The paper develops a polytopal description for quiver multiplicities in -complete bipartite quivers by gluing Knutson–Tao hive polytopes, yielding a polytope whose lattice points equal . Using quiver exceptional sequences and Derksen–Weyman embedding, the authors express as a sum of products of Littlewood–Richardson coefficients, culminating in an explicit combinatorial linear program whose feasibility governs positivity. This polytopal model, together with the Saturation Theorem, allows the membership problem for the quiver moment cone to be solved in strongly polynomial time via Tardos’ algorithm, and it provides a constructive pathway to generalizations via Vergne–Walter. The work connects representation-theoretic multiplicities, invariant theory, and polyhedral geometry, yielding both concrete computation tools and theoretical insights into quiver semi-invariants and their stability criteria.

Abstract

Let be a bipartite quiver with vertex set such that the number of arrows between any source vertex and any sink vertex is constant. Let be a dimension vector of with positive integer coordinates. Let be the representation space of -dimensional representations of and the base change group acting on be simultaneous conjugation. Let be the multiplicity of the irreducible representation of of highest weight in the ring of polynomial functions on . We show that 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 in strongly polynomial time.
Paper Structure (15 sections, 15 theorems, 89 equations, 1 figure)

This paper contains 15 sections, 15 theorems, 89 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Our results
  4. Background on Quiver Invariant Theory
  5. Quivers and their representations
  6. Weight spaces of semi-invariants and quiver semi-stability
  7. The cone of effective weights
  8. Quiver exceptional sequences and the Embedding Theorem for quiver semi-invariants
  9. Hive-type polytopes for quiver multiplicities
  10. The irreducible representations of the general linear group
  11. Knutson-Tao's hive polytopes for Littlewood-Richardson coefficients
  12. Computing weight spaces of semi-invariants via Littlewood-Richardson coefficients
  13. Hive-type polytopes for quiver multiplicities
  14. The polytope $\mathcal{P}_{\underline{\lambda}}$ from Theorem
  15. Moment cones for quivers and the proof of Theorem

Key Result

Theorem 1.1

Let $\mathbf{\mathcal{Q}}$ be an $n$-complete bipartite quiver with source vertices $x_1, \ldots, x_m$ and sink vertices $y_1, \ldots, y_\ell$ and let $\beta=(\beta(x))_{x \in \mathbf{\mathcal{Q}}_0}$ be a sincere dimension vector of $\mathbf{\mathcal{Q}}$. Let $\underline{\lambda}=(\lambda(x_i), -\

Figures (1)

  • Figure 1: Left: The $3$-hive with border labels. Right: Boundary labels determined by partitions $\lambda, \mu, \nu$.

Theorems & Definitions (43)

  • Theorem 1.1
  • Remark 2.1
  • Example 2.2
  • Theorem 2.3
  • Remark 2.4
  • Proposition 2.5
  • Remark 2.6
  • Remark 2.7
  • Lemma 2.8
  • proof
  • ...and 33 more