Journal of Lie Theory

no. 3
Graded Multiplicity in Harmonic Polynomials from the Vinberg Setting
Journal of Lie Theory, Volume 34 (2024) no. 3, pp. 677-692
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.
DOI: 10.5802/jolt.1355
Classification: 20G05
Keywords: Harmonic polynomials, theta-groups, Vinberg pair 
@article{JOLT_2024_34_3_a9,
     author = {A. Heaton},
     title = {Graded {Multiplicity} in {Harmonic} {Polynomials} from the {Vinberg} {Setting}},
     journal = {Journal of Lie Theory},
     pages = {677--692},
     year = {2024},
     volume = {34},
     number = {3},
     doi = {10.5802/jolt.1355},
     zbl = {1552.20197},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1355/}
}
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%J Journal of Lie Theory
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A. Heaton. Graded Multiplicity in Harmonic Polynomials from the Vinberg Setting. Journal of Lie Theory, Volume 34 (2024) no. 3, pp. 677-692. doi: 10.5802/jolt.1355

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