Highest weight vectors of mixed tensor products of general linear Lie superalgebras

2014/02/06 by Hebing Rui, Rui, Hebing, Yucai Su +1
Mathematics · #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.QA #math.RA #math.RT

paper · pdf · doi:10.48550/arxiv.1402.1221

arxiv created 2014/02/06 · arxiv updated 2014/02/07

Abstract

In this paper, a notion of cyclotomic (or level k) walled Brauer algebras \mathscr B_{k, r, t} is introduced for arbitrary positive integer k. It is proven that \mathscr B_{k, r, t} is free over a commutative ring with rank k^r+t(r+t)! if and only if it is admissible. Using super Schur-Weyl duality between general linear Lie superalgebras \mathfrakgl_m|n and \mathscr B_{2, r, t}, we give a classification of highest weight vectors of \mathfrakgl_m|n-modules M_pq^rt, the tensor products of Kac-modules with mixed tensor products of the natural module and its dual. This enables us to establish an explicit relationship between \mathfrakgl_m|n-Kac-modules and right cell (or standard) \mathscr B_{2, r, t}-modules over \mathbb C. Further, we find an explicit relationship between indecomposable tilting \mathfrakgl_m|n-modules appearing in M_pq^rt, and principal indecomposable right \mathscr B_{2, r, t}-modules via the notion of Kleshchev bipartitions. As an application, decomposition numbers of \mathscr B_{2, r, t} arising from super Schur-Weyl duality are determined.

Related

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• Affine walled Brauer algebras
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• Representations of cyclotomic oriented Brauer categories