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

Research output: Contribution to journalArticle

Abstract

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).

Original languageEnglish (US)
JournalJournal of Algebraic Combinatorics
DOIs
StateAccepted/In press - Jan 1 2019

Fingerprint

Superalgebra
Exterior Algebra
Module
Algebraically closed
Tensor Product
Explicit Formula
Polynomial
Zero
Arbitrary

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Discrete Mathematics and Combinatorics

Cite this

@article{d9c46aba5a864e45855ce52c696cb155,
title = "Even-primitive vectors in induced supermodules for general linear supergroups and in costandard supermodules for Schur superalgebras",
abstract = "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).",
author = "Frantisek Marko",
year = "2019",
month = "1",
day = "1",
doi = "10.1007/s10801-019-00879-6",
language = "English (US)",
journal = "Journal of Algebraic Combinatorics",
issn = "0925-9899",
publisher = "Springer Netherlands",

}

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 -