Graded multiplicity in harmonic polynomials from the Vinberg setting

TL;DR

This paper investigates harmonic polynomials associated with Vinberg $ heta$-groups from cyclic quivers, providing explicit decompositions into irreducible representations and counting multiplicities via polyhedral combinatorics.

Contribution

It offers a detailed decomposition of harmonic polynomials for specific Vinberg $ heta$-groups and introduces a combinatorial method to determine multiplicities.

Findings

Explicit irreducible decomposition of harmonic polynomials.

Counting multiplicities using integral points on polyhedral faces.

Analysis specific to the case where all $V_i$ have dimension 2.

Abstract

We consider Vinberg $\theta$-groups associated to a cyclic quiver on $r$ nodes. Let $K$ be the product of general linear groups associated to the nodes, acting naturally on $V = \oplus \text{Hom}(V_i, V_{i+1})$. We study the harmonic polynomials on $V$ in the specific case where $\dim V_i = 2$ for all $i$. For each multigraded component of the harmonics, we give an explicit decomposition into irreducible representations of $K$, and additionally describe the multiplicities of each irreducible by counting integral points on certain faces of a polyhedron.