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

TL;DR

This paper classifies highest weight vectors in tensor products of general linear Lie superalgebra modules using a new algebraic framework, establishing explicit links between superalgebra modules and certain diagrammatic algebras.

Contribution

It introduces cyclotomic walled Brauer algebras and connects their representation theory with that of general linear Lie superalgebras via super Schur-Weyl duality, providing explicit classifications and decomposition numbers.

Findings

Classification of highest weight vectors in superalgebra modules.

Explicit relationships between superalgebra modules and Brauer algebra modules.

Determination of decomposition numbers for Brauer algebras.

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 $\mathfrak{gl}_{m|n}$ and $\mathscr B_{2, r, t}$, 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) $\mathscr B_{2, r, t}$-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 $\mathscr B_{2, r, t}$-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.