Dominant and global dimension of blocks of quantised Schur algebras

Ming Fang,Steffen Koenig,Wei Hu

doi:10.1007/s00209-021-02792-w

Abstract

Group algebras of symmetric groups and their Hecke algebras are in Schur-Weyl duality with classical and quantised Schur algebras, respectively. Two homological dimensions, the dominant dimension and the global dimension, of the indecomposable summands (blocks) of these Schur algebras S(n, r) and S_q(n,r) with n geqslant r are determined explicitly, using a result on derived invariance in Fang, Hu and Koenig (J Reine Angew Math 770:59–85, 2021).

Highlights

Schur algebras have been fundamental objects in representation theory since its early days
In 1901, Schur proved in his thesis what in modern terms is called an equivalence of categories, between the polynomial representations of the general linear group G Ln(k) over an infinite field k of characteristic zero, and representations of symmetric groups r where r varies and the relevant partitions do not have more than n parts
While classical Schur algebras are crucial in the representation theory of general linear groups in describing characteristic, their quantised versions, the quantised Schur algebras introduced by Dipper and James [5], are important tools for representation theory of finite general linear groups in non-describing characteristic

Summary

Introduction

Schur algebras have been fundamental objects in representation theory since its early days. Schur-Weyl duality gives a fundamental connection between classical or quantised general linear groups and symmetric groups or Hecke algebras The strength of this connection is measured by the dominant dimension of the Schur algebra; this has been determined in [13,16] where the relevance of the dominant dimension in this context is made precise. In the classical case, when q = 1 and p > 0, a non-simple block Bτ,w has global dimension 2( pw − αp(w)) and dominant dimension 2( p − 1). Determining the global or dominant dimension of Schur algebras S(n, r ) with n < r still is an open problem, and even less is known about these dimensions of the blocks. Throughout this paper, all algebras are finite dimensional algebras over a fixed field k

Dominant dimension and global dimension

Hecke algebras and Schur algebras

Dualities