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

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) × GL(n) be its even subsupergroup. The induced supermodule HG0(λ), corresponding to a dominant weight λ of G, can be represented as HGev0(λ)⊗Λ(Y), where Y=Vm∗⊗Vn is a tensor product of the dual of the natural GL(m)-module V_m and the natural GL(n)-module V_n, and Λ (Y) is the exterior algebra of Y. For a dominant weight λ of G, we construct explicit G_ev-primitive vectors in HG0(λ). Related to this, we give explicit formulas for G_ev-primitive vectors of the supermodules HGev0(λ)⊗⊗kY. Finally, we describe a basis of G_ev-primitive vectors in the largest polynomial subsupermodule ∇ (λ) of HG0(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-primitive vectors in arbitrary induced supermodule HG0(λ).