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

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 .