Schur algebras of Brauer algebras. I.
type: Item
instance of: scholarly article
title: Schur algebras of Brauer algebras. I.
zbMATH Open document ID: 1279.20063
published in: Mathematische Zeitschrift
publication date: 2012-12-19T00:00:00Z
review text: The Schur algebra of the symmetric group is a quasi-hereditary algebra related to the group algebra of the symmetric group by Schur-Weyl duality; it is defined as the endomorphism algebra of a certain direct sum of permutation modules for the symmetric group. Numerous generalisations and variations of the Schur algebra to other settings have been defined, and in the paper under review the authors provide another variation, defining Schur algebras of Brauer algebras. The Brauer algebras are diagram algebras playing a crucial role in the representation theory of symplectic and orthogonal groups, and have also been studied in connection with mathematical physics. In fact, the existence of the Schur algebra of the Brauer algebra has already been demonstrated by the authors (with Hartmann and Paget) via abstract Schur-Weyl duality. The idea in the present paper is to provide a concrete construction emulating the construction of the classical Schur algebra: the Schur algebra of the Brauer algebra is constructed as the endomorphism ring of a direct sum of ``permutation modules'' (as defined by Hartmann and Paget). Using this construction, the authors give an explicit basis for the Schur algebra of the Brauer, which shows that it is integrally defined; they also show that it is quasi-hereditary (as one would expect for a Schur algebra); in fact, subject to certain conditions on the parameters, the new Schur algebra is a quasi-hereditary cover (as defined by Rouquier) of the Brauer algebra. The authors also show that the Schur algebra of the Brauer is built up from classical Schur algebras (just as the Brauer algebras are built up from symmetric group algebras). This is a very interesting paper, building some important theory and showing that the Schur algebra of the Brauer is worthy of further study. As one would expect from these authors, the paper is very carefully written. The only potential criticism is that the authors have apparently not followed the modern convention…
reviewed by: Matthew Fayers
Mathematics Subject Classification ID: 20G43
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 20C08
zbMATH DE Number: 6116876
zbMATH Keywords: Brauer algebras
zbMATH Keywords: Schur algebras
zbMATH Keywords: quasi-hereditary algebras
zbMATH Keywords: quasi-hereditary covers
zbMATH Keywords: Schur-Weyl duality
zbMATH Keywords: group algebras of symmetric groups
zbMATH Keywords: endomorphism algebras
zbMATH Keywords: permutation modules
author: Anne Henke
author: Steffen Koenig
MaRDI profile type: Publication
full work available at URL: s00209-011-0956-x
OpenAlex ID: W2067396655
cites work: The semisimplicity of \(\omega_f^n\)
cites work: Integral and graded quasi-hereditary algebras. I
cites work: Q5340151
cites work: Brauer algebras, symplectic Schur algebras and Schur-Weyl duality
cites work: Cohomological reduction by split pairs.
cites work: Every Semiprimary Ring is the Endomorphism Ring of a Projective Module Over a Quasi-Hereditary Ring
cites work: Q3832703
cites work: Schur-Weyl duality for orthogonal groups
cites work: Schur functors and dominant dimension
cites work: Cellular algebras
cites work: Polynomial representations of \(GL_n\)
cites work: Cohomological stratification of diagram algebras.
cites work: Young modules and filtration multiplicities for Brauer algebras.
cites work: Specht Filtrations for Hecke Algebras of Type A
cites work: Brauer algebras with parameter \(n=2\) acting on tensor space.
cites work: A characteristic free approach to Brauer algebras
cites work: Realizations of representations of the Brauer semigroup
cites work: Q3600688
DOI: 10.1007/S00209-011-0956-X
description: scientific article; zbMATH DE number 6116876