Graded multiplicity in harmonic polynomials from the Vinberg setting
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866917821944954880 |
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| author | Heaton, Alexander |
| author_facet | Heaton, Alexander |
| contents | We consider Vinberg $θ$-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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1805_03178 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Graded multiplicity in harmonic polynomials from the Vinberg setting Heaton, Alexander Representation Theory 20G05 We consider Vinberg $θ$-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. |
| title | Graded multiplicity in harmonic polynomials from the Vinberg setting |
| topic | Representation Theory 20G05 |
| url | https://arxiv.org/abs/1805.03178 |