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

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 $H_G^0(\lambda )$, corresponding to a dominant weight $\lambda$ of $G$, can be represented as $H_{G_{ev}}^0(\lambda )\otimes \Lambda (Y)$, where $Y=V_m^{\ast }\otimes V_n$ 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 $H_G^0(\lambda )$. Related to this, we give explicit formulas for $G_{ev}$-primitive vectors of the supermodules $H_{G_{ev}}^0(\lambda )\otimes {\otimes }^{k}Y$. Finally, we describe a basis of $G_{ev}$-primitive vectors in the largest polynomial subsupermodule $\nabla (\lambda )$ of $H_G^0(\lambda )$ (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 $H_G^0(\lambda )$.