Noncommutative invariants of finite and classical groups
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915687290634240 |
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| author | Ganapathy, Karthik |
| author_facet | Ganapathy, Karthik |
| contents | We investigate the structure of the invariant subring of the tensor algebra $T(W)$ of a $G$-representation $W$, viewed as a twisted commutative algebra (tca). For a faithful representation $W$ of a finite group $G$ over a field $k$, we show that if char$(k) \mid \#G$, then $T(W)^G$ is not finitely generated as a tca. In contrast, for a representation $W$ of a classical group $G_{\mathbb{Z}}$, we prove that the invariant subring $T(W_k)^{G_k}$ is finitely generated as a tca when $k$ is algebraically closed of sufficiently large characteristic, provided that $W$ admits a good filtration over $\mathbb{Z}$. Finally, we introduce a categorical variant of the Gelfand--Kirillov dimension and compute its value to be $\binom{n+1}{2}$ for $T(\mathbb{C}^n)$ as a tca. Our key insight is to use the Schur functor to reduce questions about noncommutative invariants to those concerning vector invariants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_16675 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Noncommutative invariants of finite and classical groups Ganapathy, Karthik Rings and Algebras Commutative Algebra Representation Theory 16W22 (Primary), 13A50, 16P90, 05E05 We investigate the structure of the invariant subring of the tensor algebra $T(W)$ of a $G$-representation $W$, viewed as a twisted commutative algebra (tca). For a faithful representation $W$ of a finite group $G$ over a field $k$, we show that if char$(k) \mid \#G$, then $T(W)^G$ is not finitely generated as a tca. In contrast, for a representation $W$ of a classical group $G_{\mathbb{Z}}$, we prove that the invariant subring $T(W_k)^{G_k}$ is finitely generated as a tca when $k$ is algebraically closed of sufficiently large characteristic, provided that $W$ admits a good filtration over $\mathbb{Z}$. Finally, we introduce a categorical variant of the Gelfand--Kirillov dimension and compute its value to be $\binom{n+1}{2}$ for $T(\mathbb{C}^n)$ as a tca. Our key insight is to use the Schur functor to reduce questions about noncommutative invariants to those concerning vector invariants. |
| title | Noncommutative invariants of finite and classical groups |
| topic | Rings and Algebras Commutative Algebra Representation Theory 16W22 (Primary), 13A50, 16P90, 05E05 |
| url | https://arxiv.org/abs/2502.16675 |