
Journal of Lie Theory 34 (2024), No. 3, 677692 Copyright Heldermann Verlag 2024 Graded Multiplicity in Harmonic Polynomials from the Vinberg Setting Alexander Heaton Lawrence University, Appleton, Wisconsin, U.S.A. heatona@lawrence.edu [Abstractpdf] 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. Keywords: Harmonic polynomials, thetagroups, Vinberg pair. MSC: 20G05. [ Fulltextpdf (664 KB)] for subscribers only. 