Local and global methods in representations of Hecke algebras

TL;DR

This paper develops a local-global approach for finite dimensional algebras related to Hecke algebras, proving properties like quasi-heredity and stratification, and applies these to modular representation theory and generalized q-Schur algebras.

Contribution

It introduces a local-global method for analyzing properties of algebras related to Hecke algebras, with applications to representation theory.

Findings

Many global properties hold if and only if they hold locally at each prime.

Constructed generalized q-Schur algebras are often quasi-hereditary at good primes.

Proved a triangular decomposition matrix theorem for modular Hecke algebra representations.

Abstract

This paper aims at developing a "local--global" approach for various types of finite dimensional algebras, especially those related to Hecke algebras. The eventual intention is to apply the methods and applications developed here to the cross-characteristic representation theory of finite groups of Lie type. The authors first review the notions of quasi-hereditary and stratified algebras over a Noetherian commutative ring. They prove that many global properties of these algebras hold if and only if they hold locally at every prime ideal. When the commutative ring is sufficiently good, it is often sufficient to check just the prime ideals of height at most one. These methods are applied to construct certain generalized q-Schur algebras, proving they are often quasi-hereditary (the "good" prime case) but always stratified. Finally, these results are used to prove a triangular decomposition matrix theorem for the modular representations of Hecke algebras at good primes. In the bad prime case, the generalized q-Schur algebras are at least stratified, and a block triangular analogue of the good prime case is proved, where the blocks correspond to Kazhdan-Lusztig cells.