Homological dimensions of Schur algebras $S(p,2p)$ and an Auslander-type correspondence
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2025
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| author | Cruz, Tiago Erdmann, Karin |
| author_facet | Cruz, Tiago Erdmann, Karin |
| contents | We study the homological properties of Schur algebras $S(p, 2p)$ over a field $k$ of positive characteristic $p$, focusing on their interplay with the representation theory of quotients of group algebras of symmetric groups via Schur-Weyl duality. Schur-Weyl duality establishes that the centraliser algebra, $Λ(p, 2p)$, of the tensor space $(k^p)^{\otimes 2p}$ (as a module over $S(p, 2p)$) is a quotient of the group algebra of the symmetric group. In this paper, we prove that Schur-Weyl duality between $S(p, 2p)$ and $Λ(p, 2p)$ is an instance of an Auslander-type correspondence.
We compute the global dimension of Schur algebras $S(p, 2p)$ and their relative dominant dimension with respect to the tensor space $(k^p)^{\otimes 2p}$. In particular, we show that the pair $(S(p, 2p), (k^p)^{\otimes 2p})$ forms a relative $4(p-1)$-Auslander pair in the sense of Cruz and Psaroudakis, thereby connecting Schur algebras with higher homological algebra. Moreover, we determine the Hemmer-Nakano dimension associated with the quasi-hereditary cover of $Λ(p, 2p)$ that arises from Schur-Weyl duality. As an application, we show that the direct sum of some Young modules over $Λ(p, 2p)$ is a full tilting module when $p>2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_04460 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homological dimensions of Schur algebras $S(p,2p)$ and an Auslander-type correspondence Cruz, Tiago Erdmann, Karin Representation Theory 16E10, 20G43 (Primary) 16G10, 20C30, 16E65, 18G25 (Secondary) We study the homological properties of Schur algebras $S(p, 2p)$ over a field $k$ of positive characteristic $p$, focusing on their interplay with the representation theory of quotients of group algebras of symmetric groups via Schur-Weyl duality. Schur-Weyl duality establishes that the centraliser algebra, $Λ(p, 2p)$, of the tensor space $(k^p)^{\otimes 2p}$ (as a module over $S(p, 2p)$) is a quotient of the group algebra of the symmetric group. In this paper, we prove that Schur-Weyl duality between $S(p, 2p)$ and $Λ(p, 2p)$ is an instance of an Auslander-type correspondence. We compute the global dimension of Schur algebras $S(p, 2p)$ and their relative dominant dimension with respect to the tensor space $(k^p)^{\otimes 2p}$. In particular, we show that the pair $(S(p, 2p), (k^p)^{\otimes 2p})$ forms a relative $4(p-1)$-Auslander pair in the sense of Cruz and Psaroudakis, thereby connecting Schur algebras with higher homological algebra. Moreover, we determine the Hemmer-Nakano dimension associated with the quasi-hereditary cover of $Λ(p, 2p)$ that arises from Schur-Weyl duality. As an application, we show that the direct sum of some Young modules over $Λ(p, 2p)$ is a full tilting module when $p>2$. |
| title | Homological dimensions of Schur algebras $S(p,2p)$ and an Auslander-type correspondence |
| topic | Representation Theory 16E10, 20G43 (Primary) 16G10, 20C30, 16E65, 18G25 (Secondary) |
| url | https://arxiv.org/abs/2507.04460 |