A characteristic free approach to Brauer algebras (Q2701667)

The Brauer algebra \(B_k(r,\delta)\) is an associative algebra depending on a natural number \(r\), a field \(k\) and a parameter \(\delta\in k\). Brauer algebras first arose in the orthogonal and symplectic analogues of Schur-Weyl duality, and have also featured in the context of quantum groups and low-dimensional topology.NEWLINENEWLINENEWLINEThe irreducible modules of the Brauer algebra in the case \(k=\mathbb{C}\) are well understood thanks to work of Brown, Hanlon, Wales, Wenzl and Kerov. The focus of the paper under review is instead the ring structure of the Brauer algebra, which is considerably less well understood. The methods used build on the authors' equivalent definition [given in CMS Conf. Proc. 24, 365-386 (1998; Zbl 0926.16016)] of \textit{J. J. Graham} and \textit{G. I. Lehrer}'s cellular algebras [Invent. Math. 123, No. 1, 1-34 (1996; Zbl 0853.20029)]. In [J. Lond. Math. Soc., II. Ser. 60, No. 3, 700-722 (1999; Zbl 0952.16014)], the authors showed how to construct all cellular algebras (and no others) using a technique called inflation. This paper gives a detailed construction of the Brauer algebra using this approach, and shows how Graham and Lehrer's results on the Brauer algebra may be recovered from it.NEWLINENEWLINENEWLINEIn \S 7, the authors consider the blocks of the Brauer algebra in the case that the parameter \(\delta\) is nonsingular, which only excludes finitely many possible values. They obtain theorems about the ring structure of the algebra and a necessary and sufficient criterion for a block of the algebra to be of finite representation type.