A combinatorial approach to Donkin-Koppinen filtrations of general linear supergroups

For a general linear supergroup $G=GL(m|n)$, we consider a natural isomorphism $ϕ: G \to U^-\times G_{ev} \times U^+$, where $G_{ev}$ is the even subsupergroup of $G$, and $U^-$, $U^+$ are appropriate odd unipotent subsupergroups of $G$. We compute the action of odd superderivations on the images $ϕ^*(x_{ij})$ of the generators of $K[G]$. We describe a specific ordering of the dominant weights $X(T)^+$ of $GL(m|n)$ for which there exists a Donkin-Koppinen filtration of the coordinate algebra $K[G]$. Let $Γ$ be a finitely generated ideal $Γ$ of $X(T)^+$ and $O_Γ(K[G])$ be the largest $Γ$-subsupermodule of $K[G]$ having simple composition factors of highest weights $λ\in Γ$. We apply combinatorial techniques, using generalized bideterminants, to determine a basis of $G$-superbimodules appearing in Donkin-Koppinen filtration of $O_Γ(K[G])$.

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