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 ${\mathcal{B}}_{k,r,t}$ is introduced for arbitrary positive integer $k$. It is proven that ${\mathcal{B}}_{k,r,t}$ 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 ${\mathfrak{gl}}_{m|n}$ and ${\mathcal{B}}_{2,r,t}$, we give a classification of highest weight vectors of ${\mathfrak{gl}}_{m|n}$-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 ${\mathfrak{gl}}_{m|n}$-Kac-modules and right cell (or standard) ${\mathcal{B}}_{2,r,t}$-modules over $\mathbb{C}$. Further, we find an explicit relationship between indecomposable tilting ${\mathfrak{gl}}_{m|n}$-modules appearing in $M_{pq}^{rt}$, and principal indecomposable right ${\mathcal{B}}_{2,r,t}$-modules via the notion of Kleshchev bipartitions. As an application, decomposition numbers of ${\mathcal{B}}_{2,r,t}$ arising from super Schur-Weyl duality are determined.

References in corpus (1)

• Affine walled Brauer algebras

Cited by in corpus (1)

• A basis theorem for the affine oriented Brauer category and its cyclotomic quotients