$τ$-Tilting finiteness of group algebras over generalized symmetric groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913353989881856 |
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| author | Hiramae, Naoya |
| author_facet | Hiramae, Naoya |
| contents | In this paper, we show that weakly symmetric $τ$-tilting finite algebras have positive definite Cartan matrices, which implies that we can prove $τ$-tilting infiniteness of weakly symmetric algebras by calculating their Cartan matrices. Similarly, we obtain the condition on Cartan matrices that selfinjective algebras are $τ$-tilting infinite. By applying this result, we show that a group algebra of $(\mathbb{Z}/p^l\mathbb{Z})^n\rtimes H$ is $τ$-tilting infinite when $p^l\geq n$ and $\#\mathrm{IBr}\,H\geq\min\{p,3\}$, where $p>0$ is the characteristic of the ground field, $H$ is a subgroup of the symmetric group $\mathfrak{S}_n$ of degree $n$, the action of $H$ permutes the entries of $(\mathbb{Z}/p^l\mathbb{Z})^n$, and $\mathrm{IBr}\,H$ denotes the set of irreducible $p$-Brauer characters of $H$. Moreover, we show that under the assumption that $p^l\geq n$ and $H$ is a $p'$-subgroup of $\mathfrak{S}_n$, $τ$-tilting finiteness of a group algebra of a group $(\mathbb{Z}/p^l\mathbb{Z})^n\rtimes H$ is determined by its $p$-hyperfocal subgroup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_10726 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $τ$-Tilting finiteness of group algebras over generalized symmetric groups Hiramae, Naoya Representation Theory Group Theory Rings and Algebras 16G10, 20C20 In this paper, we show that weakly symmetric $τ$-tilting finite algebras have positive definite Cartan matrices, which implies that we can prove $τ$-tilting infiniteness of weakly symmetric algebras by calculating their Cartan matrices. Similarly, we obtain the condition on Cartan matrices that selfinjective algebras are $τ$-tilting infinite. By applying this result, we show that a group algebra of $(\mathbb{Z}/p^l\mathbb{Z})^n\rtimes H$ is $τ$-tilting infinite when $p^l\geq n$ and $\#\mathrm{IBr}\,H\geq\min\{p,3\}$, where $p>0$ is the characteristic of the ground field, $H$ is a subgroup of the symmetric group $\mathfrak{S}_n$ of degree $n$, the action of $H$ permutes the entries of $(\mathbb{Z}/p^l\mathbb{Z})^n$, and $\mathrm{IBr}\,H$ denotes the set of irreducible $p$-Brauer characters of $H$. Moreover, we show that under the assumption that $p^l\geq n$ and $H$ is a $p'$-subgroup of $\mathfrak{S}_n$, $τ$-tilting finiteness of a group algebra of a group $(\mathbb{Z}/p^l\mathbb{Z})^n\rtimes H$ is determined by its $p$-hyperfocal subgroup. |
| title | $τ$-Tilting finiteness of group algebras over generalized symmetric groups |
| topic | Representation Theory Group Theory Rings and Algebras 16G10, 20C20 |
| url | https://arxiv.org/abs/2405.10726 |