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 $$H^0_G(\lambda )$$, 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 $$H^0_G(\lambda )$$.