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

2016/08/31 by Frantisek Marko, Marko, Frantisek
Mathematics · #FOS: Mathematics #Representation Theory (math.RT) #math.RT

paper · pdf · doi:10.48550/arxiv.1608.08989

arxiv created 2017/08/06 · arxiv updated 2017/08/08

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)× GL(n) be its even subsupergroup. The induced supermodule H⁰_G(λ), corresponding to a dominant weight λ of G, can be represented as H⁰_G_ev(λ)⊗ Λ(Y), where Y=V_m^*⊗ 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 Λ(Y) is the exterior algebra of Y. For a dominant weight λ of G, we construct explicit G_ev-primitive vectors in H⁰_G(λ). Related to this, we give explicit formulas for G_ev-primitive vectors of the supermodules H⁰_G_ev(λ)⊗ ⊗^k Y. Finally, we describe a basis of G_ev-primitive vectors in the largest polynomial subsupermodule ∇(λ) of H⁰_G(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 H⁰_G(λ).

Related

• Primitive vectors in induced supermodules for general linear supergroups
• Combinatorial aspects of the linkage principle for general linear supergroups
• Levi-type Schur-Sergeev duality for general linear super groups
• Polynomial super representations of the hyperalgebra of \mathfrakgl_m|n at roots of unity
• Irreducibility of induced modules for general linear supergroups