Abstract
In this paper, a notion of cyclotomic (or level k) walled Brauer algebras ℬ k, r, t is introduced for arbitrary positive integer k. It is proven that 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 \( \mathfrak{g}{\mathfrak{l}}_{m\left|n\right.} \) and ℬ 2, r, t , we give a classification of highest weight vectors of \( \mathfrak{g}{\mathfrak{l}}_{m\left|n\right.} \) -modules M rt pq , 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 \( \mathfrak{g}{\mathfrak{l}}_{m\left|n\right.} \) -Kac-modules and right cell (or standard) ℬ 2, r, t -modules over C. Further, we find an explicit relationship between indecomposable tilting \( \mathfrak{g}{\mathfrak{l}}_{m\left|n\right.} \) -modules appearing in M rt pq , and principal indecomposable right ℬ 2, r, t -modules via the notion of Kleshchev bipartitions. As an application, decomposition numbers of ℬ 2, r, t arising from super Schur-Weyl duality are determined.
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Supported by NSFC no. 11025104, and SMSTC no. 11XD1402200.
Supported by NSFC no. 11371278, 11431010, SMSTC no. 12XD1405000, and Fundamental Research Funds for the Central Universities of China.
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RUI, H., SU, Y. HIGHEST WEIGHT VECTORS OF MIXED TENSOR PRODUCTS OF GENERAL LINEAR LIE SUPERALGEBRAS. Transformation Groups 20, 1107–1140 (2015). https://doi.org/10.1007/s00031-015-9331-z
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DOI: https://doi.org/10.1007/s00031-015-9331-z