### 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 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).

Original language | English (US) |
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Journal | Journal of Algebraic Combinatorics |

DOIs | |

State | Accepted/In press - Jan 1 2019 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Discrete Mathematics and Combinatorics

### Cite this

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**Even-primitive vectors in induced supermodules for general linear supergroups and in costandard supermodules for Schur superalgebras.** / Marko, Frantisek.

Research output: Contribution to journal › Article

TY - JOUR

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

AU - Marko, Frantisek

PY - 2019/1/1

Y1 - 2019/1/1

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).

UR - http://www.scopus.com/inward/record.url?scp=85072020734&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85072020734&partnerID=8YFLogxK

U2 - 10.1007/s10801-019-00879-6

DO - 10.1007/s10801-019-00879-6

M3 - Article

AN - SCOPUS:85072020734

JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

SN - 0925-9899

ER -