Homological dimensions of Schur algebras $S(p,2p)$ and an Auslander-type correspondence

TL;DR

This paper investigates the homological properties of Schur algebras $S(p, 2p)$, revealing their connection to higher homological algebra through an Auslander-type correspondence and computing their global and dominant dimensions.

Contribution

It establishes an Auslander-type correspondence for Schur-Weyl duality, computes global and dominant dimensions, and links Schur algebras to higher homological algebra concepts.

Findings

Schur-Weyl duality is an instance of an Auslander-type correspondence.

Computed the global dimension of $S(p, 2p)$ and their relative dominant dimension.

Identified conditions under which certain Young modules form a full tilting module.

Abstract

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, $\Lambda(p, 2p)$, 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 $\Lambda(p, 2p)$ 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 $\Lambda(p, 2p)$ that arises from Schur-Weyl duality. As an application, we show that the direct sum of some Young modules over $\Lambda(p, 2p)$ is a full tilting module when $p>2$.