TY - JOUR

T1 - Even-primitive vectors in induced supermodules for general linear supergroups and in costandard supermodules for Schur superalgebras

AU - Marko, František

N1 - Publisher Copyright:
© 2019, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2020/5/1

Y1 - 2020/5/1

N2 - 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).

AB - 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).

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U2 - 10.1007/s10801-019-00879-6

DO - 10.1007/s10801-019-00879-6

M3 - Article

AN - SCOPUS:85072020734

SN - 0925-9899

VL - 51

SP - 369

EP - 417

JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

IS - 3

ER -