Cluster Algebras, Invariant Theory, and Kronecker Coefficients II

TL;DR

This paper establishes that the semi-invariant ring of a specific Kronecker quiver is an upper cluster algebra, providing explicit quivers and a method to compute Kronecker coefficients via lattice point counting in polytopes.

Contribution

It proves the semi-invariant ring forms an upper cluster algebra and introduces a polyhedral model for computing Kronecker coefficients.

Findings

Semi-invariant ring is an upper cluster algebra.

Explicit quiver and cluster are constructed.

Kronecker coefficients can be computed via lattice points in polytopes.

Abstract

We prove that the semi-invariant ring of the standard representation space of the $l$-flagged $m$-arrow Kronecker quiver is an upper cluster algebra for any $l,m\in \mathbb{N}$. The quiver and cluster are explicitly given. We prove that the quiver with its rigid potential is a polyhedral cluster model. As a consequence, to compute each Kronecker coefficient $g_{\mu,\nu}^\lambda$ with $\lambda$ at most $m$ parts, we only need to count lattice points in at most $m!$ fibre (rational) polytopes inside the ${\rm g}$-vector cone, which is explicitly given.