Journal of Lie Theory 34 (2024), No. 3, 677--692
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
[Abstract-pdf]
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, theta-groups, Vinberg pair.
MSC: 20G05.
[ Fulltext-pdf (664 KB)] for subscribers only.