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

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.

An action of the Hecke monoid on rational modules for the Borel subgroup of a quantised general linear group

“…As it is well known, see e.g. [8], [9], the associative algebra S − (n, r) = S − α,β (n, r) dual to A(δ; r) is called the (negative) quantised Borel-Schur algebra. As usual, we have a canonical equivalence between the categories S − (n, r)-Mod and Comod-A(δ; r).…”

“…Quantum parabolic and Borel subgroups were extensively studied by Donkin in [8] (see also [9]), for the case α = 1.…”

Stratifying ideals and twisted products

“…The aim of this paper is to give a new sufficient condition, Theorem 3.7, for an idempotent ideal ΛeΛ to be stratifying. This result will be used in our work on homological properties of (quantised) Schur algebras (see [10,6]).…”