Extensions of modules over Schur algebras, symmetric groups and Hecke algebras.
type: Item
instance of: scholarly article
title: Extensions of modules over Schur algebras, symmetric groups and Hecke algebras.
zbMATH Open document ID: 1084.20004
DOI: 10.1023/B:ALGE.0000019454.27331.59
author: Karin Erdmann
author: Daniel K. Nakano
author: S. R. Doty
published in: Algebras and Representation Theory
publication date: 2005-04-07T00:00:00Z
review text: Schur-Weyl duality gives a well-known relationship between the representation theory of finite symmetric groups (\(\Sigma_d\)) and general linear groups (\(\text{GL}_n(k)\)) over a field \(k\). The Schur algebra \(S(n,d)\) arises from this relationship and its representations are certain polynomial representations of \(\text{GL}_n(k)\). Motivated by this relationship, the authors use homological tools (specifically spectral sequences) on Schur algebras to obtain explicit relationships between extensions for symmetric groups and those for general linear groups. The spectral sequences are constructed using two adjoint functors to the Schur functor \({\mathcal F}\colon S(n,d)\to k\Sigma_d\). These adjoint functors were studied earlier by \textit{M. Auslander} [Commun. Algebra 1, 177-268 (1974; Zbl 0285.16028)]. The authors work in a more general setting by taking an arbitrary finite-dimensional algebra \(A\) over a field \(k\) with an idempotent element \(e\). Then the ``Schur functor'' maps \(A\) to \(eAe\). The adjoint functors give rise to two Grothendieck spectral sequences relating extensions over \(A\) to extensions over \(eAe\). These spectral sequences along with study of the adjoint functors allow the authors to deduce several explicit relationships between extensions. The authors then specialize to the case when \(A\) is a quasi-hereditary algebra. For example, \(A\) could be a Schur algebra or \(q\)-Schur algebra. (In the latter case, modules for the \(q\)-Schur algebra correspond to modules for a quantum general linear group and \(eAe\) is a Hecke algebra.) The general results lead to even more precise relationships involving \(\text{Ext}^1\)- and \(\text{Ext}^2\)-groups over \(A\) and \(eAe\), from which some known results are quite nicely recovered. A vanishing result and further relationships are given. Then the authors focus on the case when \(A\) is a \(q\)-Schur algebra of the form \(S_q(d,d)\) (which can be reduced to the ordinary Schur algebra \(S(d…
reviewed by: Christopher P. Bendel
Mathematics Subject Classification ID: 20C08
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 18G40
zbMATH DE Number: 2153663
zbMATH Keywords: Schur algebras
zbMATH Keywords: Hecke algebras
zbMATH Keywords: extensions
zbMATH Keywords: cohomology
zbMATH Keywords: finite-dimensional algebras
zbMATH Keywords: quasi-hereditary algebras
zbMATH Keywords: quantum general linear groups
zbMATH Keywords: Schur functors
zbMATH Keywords: spectral sequences
zbMATH Keywords: categories of modules
zbMATH Keywords: composition factor multiplicities
zbMATH Keywords: derived functors
zbMATH Keywords: Young modules
zbMATH Keywords: symmetric groups
MaRDI profile type: Publication
full work available at URL: b:alge.0000019454.27331.59
OpenAlex ID: W2042175345
description: scientific article; zbMATH DE number 2153663