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

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 C_Γ = O_Γ(K[G]). Here, the supermodule C_Γ 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^–×G_ev×U⁺ induces a superalgebra isomorphism ϕ^* K[U^–]⊗K[G_ev]⊗K[U⁺]≃K[G]. We show that C_Γ=ϕ^*(K[U^–]⊗M_ΓK[U⁺]), where M_Γ=O_Γ(K[G_ev]). Using the basis of the module M_Γ, given by generalized bideterminants, we describe a basis of C_Γ.

Since each C_Γ is a subsupercoalgebra of K[G], its dual ${C}_{\Gamma}^{\ast }={S}_{\Gamma}$ is a (pseudocompact) superalgebra called the generalized Schur superalgebra. There is a natural superalgebra morphism π_Γ : Dist(G) → S_Γ such that the image of the distribution algebra Dist(G) is dense in S_Γ. For the ideal $X{(T)}_l^{+},$ of all weights of fixed length l, the generators of the kernel of ${\uppi}_{X{(T)}_l^{+}}$ are described.