Stable Graded Multiplicities for Harmonics on a Cyclic Quiver
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909120086409216 |
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| author | Frohmader, Andrew Heaton, Alexander |
| author_facet | Frohmader, Andrew Heaton, Alexander |
| contents | We consider Vinberg $θ$-groups associated to a cyclic quiver on $k$ nodes. Let $K$ be the product of the general linear groups associated to each node. Then $K$ acts naturally on $\oplus \text{Hom}(V_i, V_{i+1})$ and by Vinberg's theory the polynomials are free over the invariants. We therefore consider the harmonics as a representation of $K$, and give a combinatorial formula for the stable graded multiplicity of each $K$-type. A key lemma provides a combinatorial separation of variables that allows us to cancel the invariants and obtain generalized exponents for the harmonics. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_16198 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stable Graded Multiplicities for Harmonics on a Cyclic Quiver Frohmader, Andrew Heaton, Alexander Representation Theory Combinatorics We consider Vinberg $θ$-groups associated to a cyclic quiver on $k$ nodes. Let $K$ be the product of the general linear groups associated to each node. Then $K$ acts naturally on $\oplus \text{Hom}(V_i, V_{i+1})$ and by Vinberg's theory the polynomials are free over the invariants. We therefore consider the harmonics as a representation of $K$, and give a combinatorial formula for the stable graded multiplicity of each $K$-type. A key lemma provides a combinatorial separation of variables that allows us to cancel the invariants and obtain generalized exponents for the harmonics. |
| title | Stable Graded Multiplicities for Harmonics on a Cyclic Quiver |
| topic | Representation Theory Combinatorics |
| url | https://arxiv.org/abs/2402.16198 |