Let G= GL(m| n) be the general linear supergroup over an algebraically closed field K of characteristic zero, and let Gev= 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 Vm and the natural GL(n)-module Vn, and Λ (Y) is the exterior algebra of Y. For a dominant weight λ of G, we construct explicit Gev-primitive vectors in HG0(λ). Related to this, we give explicit formulas for Gev-primitive vectors of the supermodules HGev0(λ)⊗⊗kY. Finally, we describe a basis of Gev-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 Gev-primitive vectors in arbitrary induced supermodule HG0(λ).
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Discrete Mathematics and Combinatorics