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The blocks of the Brauer algebra in characteristic zero. (English) Zbl 1230.20046

From the introduction: We determine the blocks of the Brauer algebra in characteristic zero. We also give information on the submodule structure of standard modules for this algebra.
The Brauer algebra \(B_n(\delta)\) was introduced by R. Brauer [in Ann. Math. (2) 38, 857-872 (1937; Zbl 0017.39105)] in the study of the representation theory of orthogonal and symplectic groups. Over \(\mathbb C\), and for integral values of \(\delta\), its action on tensor space \(T=(\mathbb C^{|\delta|})^{\otimes n}\) can be identified with the centraliser algebra for the corresponding group action. This generalises the Schur-Weyl duality between symmetric and general linear groups [H. Weyl, The classical groups, their invariants and representations. Princeton University Press (1939; Zbl 0020.20601)].
The paper begins with a section defining the various objects of interest, and a review of their basic properties in the spirit of A. Cox, P. Martin, A. Parker, and C. Xi [J. Algebra 302, No. 1, 340-360 (2006; Zbl 1147.16016)]. This is followed by a brief section describing some basic results about Littlewood-Richardson coefficients which will be needed in what follows. In Section 4 we begin the analysis of blocks by giving a necessary condition for two weights to be in the same block. This is based on an analysis of the action of certain central elements in the algebra on standard modules, and inductive arguments using Frobenius reciprocity. Section 5 constructs homomorphisms between standard modules in certain special cases, generalising a result of W. F. Doran IV, D. B. Wales, and P. J. Hanlon [in J. Algebra 211, No. 2, 647-685 (1999; Zbl 0944.16002)]. Although not necessary for the main block result, this is of independent interest.
The classification of blocks is completed in Section 6. The main idea is to show that every block contains a unique minimal weight, and that there is a homomorphism from any standard labelled by a non-minimal weight to one labelled by a smaller weight. We also describe precisely which weights are minimal in their blocks.
In Section 7 we consider a certain explicit choice of weights, and show inductively, via Frobenius reciprocity arguments, that the corresponding standards can have arbitrarily complicated submodule structures. We conclude by outlining the modifications to our arguments required in the case \(\delta=0\).

MSC:

20G05 Representation theory for linear algebraic groups
20C08 Hecke algebras and their representations
20C30 Representations of finite symmetric groups
05E10 Combinatorial aspects of representation theory
16G20 Representations of quivers and partially ordered sets
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