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

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