HIGHEST WEIGHT VECTORS OF MIXED TENSOR PRODUCTS OF GENERAL LINEAR LIE SUPERALGEBRAS

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 , 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 , 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.