Schur algebras of Brauer algebras. I.
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 of providing keywords or subject classifications for the paper (but perhaps this is the convention of the journal).
scientific article; zbMATH DE number 6116876
scientific article