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

2025/07/06 by Tiago Cruz, Cruz, Tiago, Karin Erdmann +1
Mathematics · #16E10 #16E65 #18G25 (Secondary) #20C30 #20G43 (Primary) 16G10 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2507.04460

openalex publication_date 2025/07/06 · openalex created_date 2025/10/20 · openalex updated_date 2026/08/01

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, Λ(p, 2p), of the tensor space (k^p)^{⊗ 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)^{⊗ 2p}. In particular, we show that the pair (S(p, 2p), (k^p)^{⊗ 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.

Citations

• A higher dimensional Auslander-Iyama-Solberg correspondence
• Characteristic tilting modules and Ringel duality in the Noetherian world
• Higher Morita-Tachikawa correspondence
• Cellularity of endomorphism algebras of tilting objects
• On Ext-Quivers of Blocks of Weight Two for Symmetric Groups
• q-Schur algebras and complex reflection groups, I
• On the good filtration dimension of Weyl modules for a linear algebraic group
• The Global Dimension of Schur Algebras for GL₂ and GL₃

Related

• Brauer algebras, symplectic Schur algebras and Schur-Weyl duality
• Young Modules and Schur algebras
• Generalized Schur Algebras
• Torsion pairs and Ringel duality for Schur algebras
• A note on Schur-Weyl dualities for GL(m) and GL(m|n)