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

  • František MarkoEmail author
Article
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Abstract

Let \(G=GL(m|n)\) be the general linear supergroup over an algebraically closed field K of characteristic zero, and let \(G_{ev}=GL(m)\times GL(n)\) be its even subsupergroup. The induced supermodule \(H^0_G(\lambda )\), corresponding to a dominant weight \(\lambda \) of G, can be represented as \(H^0_{G_{ev}}(\lambda )\otimes \Lambda (Y)\), where \(Y=V_m^*\otimes V_n\) is a tensor product of the dual of the natural GL(m)-module \(V_m\) and the natural GL(n)-module \(V_n\), and \(\Lambda (Y)\) is the exterior algebra of Y. For a dominant weight \(\lambda \) of G, we construct explicit \(G_{ev}\)-primitive vectors in \(H^0_G(\lambda )\). Related to this, we give explicit formulas for \(G_{ev}\)-primitive vectors of the supermodules \(H^0_{G_{ev}}(\lambda )\otimes \otimes ^k Y\). Finally, we describe a basis of \(G_{ev}\)-primitive vectors in the largest polynomial subsupermodule \(\nabla (\lambda )\) of \(H^0_G(\lambda )\) (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 \(H^0_G(\lambda )\).

Keywords

General linear supergroup Primitive vectors Schur superalgebra 

Mathematics Subject Classification

Primary 15A15 Secondary 17A70 20G05 15A72 13A50 05E15 

Notes

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Penn State HazletonHazletonUSA

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