Homological properties of quantised Borel-Schur algebras and resolutions of quantised Weyl modules

TL;DR

This paper advances the homological understanding of quantised Borel-Schur algebras and uses this to establish resolutions of quantised Weyl modules, linking quantum group theory with Hecke algebra representations.

Contribution

It develops the homological theory of quantum general linear groups and applies it to construct resolutions of co-Specht modules for Hecke algebras.

Findings

Acyclicity of induction from rank-one modules established

Exactness of complexes for resolutions proven

Connections made between quantum groups and Hecke algebra representations

Abstract

We continue the development of the homological theory of quantum general linear groups previously considered by the first author. The development is used to transfer information to the representation theory of quantised Schur algebras. The acyclicity of induction from some rank-one modules for quantised Borel-Schur subalgebras is deduced. This is used to prove the exactness of the complexes recently constructed by Boltje and Maisch, giving resolutions of the co-Specht modules for Hecke algebras.