Even-primitive vectors in induced supermodules for general linear supergroups and in costandard supermodules for Schur superalgebras

TL;DR

This paper constructs explicit primitive vectors in induced supermodules for general linear supergroups and describes their bases, advancing understanding of supermodule structure in representation theory.

Contribution

It provides explicit formulas and bases for primitive vectors in induced supermodules and costandard modules for Schur superalgebras, a novel detailed structural analysis.

Findings

Explicit $G_{ev}$-primitive vectors constructed for induced supermodules.

Formulas for primitive vectors in tensor products involving $Y$.

Basis description for primitive vectors in costandard supermodules.

Abstract

Let $G=GL(m|n)$ be the general linear supergroup over an algebraically closed field $K$ of characteristic zero and let $G_{ev}=GL(m)\times GL(n)$ be its even subsupergroup. The induced supermodule $H^0_G(\lambda)$, corresponding to a dominant weight $\lambda$ of $G$, can be represented as $H^0_{G_{ev}}(\lambda)\otimes \Lambda(Y)$, where $Y=V_m^*\otimes V_n$ is a tensor product of the dual of the natural $GL(m)$-module $V_m$ and the natural $GL(n)$-module $V_n$, and $\Lambda(Y)$ is the exterior algebra of $Y$. For a dominant weight $\lambda$ of $G$, we construct explicit $G_{ev}$-primitive vectors in $H^0_G(\lambda)$. Related to this, we give explicit formulas for $G_{ev}$-primitive vectors of the supermodules $H^0_{G_{ev}}(\lambda)\otimes \otimes^k Y$. Finally, we describe a basis of $G_{ev}$-primitive vectors in the largest polynomial subsupermodule $\nabla(\lambda)$ of $H^0_G(\lambda)$ (and therefore in the costandard supermodule of the corresponding Schur superalgebra $S(m|n)$). This yields a description of a basis of $G_{ev}$-primitive vectors in arbitrary induced supermodule $H^0_G(\lambda)$.