Brauer algebras with parameter \(n=2\) acting on tensor space. (English) Zbl 1185.20048
Let \(E\) be an \(n\)-dimensional vector space over a field \(k\) of prime characteristic \(p\). In this paper the authors study the \(r\)-tensor space \(E^{\otimes r}\) as a module for the Brauer algebra \(B_k(r,\delta)\) in the special case where \(\delta=2=n\). They give a decomposition of \(E^{\otimes r}\) into a direct sum of some submodules and analyse the structure of these submodules in the case \(p\neq 2\). It turns out that the tensor space still affords Schur-Weyl duality. But it is not filtered by cell modules and thus not equal to a direct sum of Young modules as defined by R. Hartmann and R. Paget [Math. Z. 254, No. 2, 333-357 (2006; Zbl 1116.16009)].
Reviewer: Hu Jun (Beijing)
MSC:
| 20G05 | Representation theory for linear algebraic groups |
| 05E10 | Combinatorial aspects of representation theory |
| 16G10 | Representations of associative Artinian rings |
Citations:
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