Trace forms on the cyclotomic Hecke algebras and cocenters of the cyclotomic Schur algebras
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866916580199235584 |
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| author | He, Zhekun Hu, Jun Lin, Huang |
| author_facet | He, Zhekun Hu, Jun Lin, Huang |
| contents | We define a unified trace form $τ$ on the cyclotomic Hecke algebras $\mathscr{H}_{n,K}$ of type $A$, which generalize both Malle-Mathas' trace form on the non-degenerate version (with Hecke parameter $ξ\neq 1$) and Brundan-Kleshchev's trace form on the degenerate version. We use seminormal basis theory to construct a pair of dual bases for $\mathscr{H}_{n,K}$ with respect to the form. We also construct an explicit basis for the cocenter (i.e., the $0$th Hochschild homology) of the corresponding cyclotomic Schur algebra, which shows that the cocenter has dimension independent of the ground field $K$, the Hecke parameter $ξ$ and the cyclotomic parameters $Q_1,\cdots,Q_\ell$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2208_05172 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Trace forms on the cyclotomic Hecke algebras and cocenters of the cyclotomic Schur algebras He, Zhekun Hu, Jun Lin, Huang Representation Theory We define a unified trace form $τ$ on the cyclotomic Hecke algebras $\mathscr{H}_{n,K}$ of type $A$, which generalize both Malle-Mathas' trace form on the non-degenerate version (with Hecke parameter $ξ\neq 1$) and Brundan-Kleshchev's trace form on the degenerate version. We use seminormal basis theory to construct a pair of dual bases for $\mathscr{H}_{n,K}$ with respect to the form. We also construct an explicit basis for the cocenter (i.e., the $0$th Hochschild homology) of the corresponding cyclotomic Schur algebra, which shows that the cocenter has dimension independent of the ground field $K$, the Hecke parameter $ξ$ and the cyclotomic parameters $Q_1,\cdots,Q_\ell$. |
| title | Trace forms on the cyclotomic Hecke algebras and cocenters of the cyclotomic Schur algebras |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2208.05172 |