DONKIN–KOPPINEN FILTRATION FOR GL(m|n) AND GENERALIZED SCHUR SUPERALGEBRAS

F. Marko, A. N. Zubkov

Research output: Contribution to journalArticlepeer-review

Abstract

The paper contains results that characterize the Donkin–Koppinen filtration of the coordinate superalgebra K[G] of the general linear supergroup G = GL(m|n) by its subsupermodules CΓ = OΓ(K[G]). Here, the supermodule CΓ is the largest subsupermodule of K[G] whose composition factors are irreducible supermodules of highest weight ⋋, where ⋋ belongs to a finitely-generated ideal Γ of the poset X(T)+ of dominant weights of G. A decomposition of G as a product of subsuperschemes U×Gev×U+ induces a superalgebra isomorphism ϕ*K[U]⊗K[Gev]⊗K[U+]≃K[G]. We show that CΓ*(K[U]⊗MΓK[U+]), where MΓ=OΓ(K[Gev]). Using the basis of the module MΓ, given by generalized bideterminants, we describe a basis of CΓ. Since each CΓ is a subsupercoalgebra of K[G], its dual CΓ∗=SΓ is a (pseudocompact) superalgebra called the generalized Schur superalgebra. There is a natural superalgebra morphism πΓ : Dist(G) → SΓ such that the image of the distribution algebra Dist(G) is dense in SΓ. For the ideal X(T)l+, of all weights of fixed length l, the generators of the kernel of πX(T)l+ are described.

Original languageEnglish
JournalTransformation Groups
DOIs
Publication statusAccepted/In press - 2022

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology

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