Irreducibility of induced supermodules for general linear supergroups

Some homological properties of GL(m|n) in arbitrary characteristic

“…Arguing as in the above theorem and using Corollary 12.4 one can suppose that B is standard and even more, B = B − id . It remains to combine Theorem 1 from [14] with Corollary II.5.6 from [10]. Proof.…”

Borel-Weil theorem for algebraic supergroups

“…On the other hand, the study of quasireductive supergroups over a field of positive characteristic has just started. A. Zubkov and F. Marko [14,15,32,33] have obtained may results for representations of Ä(m|n). J. Brundan and A. Kleshchev [3] studied representations of Q(n) which have a close relationship to modular representations of spin symmetric groups.…”

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

“…The affirmative answer to this question is the primary goal of this paper. To accomplish this, we consider a natural isomorphism φ : G → U − × G ev × U + , where G ev is the even subsupergroup of G, and U − , U + are appropriate odd unipotent subsupergroups of G. We describe an explicit ordering of the dominant weights of GL(m|n) and completely describe the action of superderivations on the images φ * (x ij ) of the generators of K[G], which is of independent interest and is an extension of the results from [7] and [8]. That way, we obtain a handson description of the Donkin-Koppinen filtration for O Γ (K[G]) for G = GL(m|n) considered initially in [9].…”

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For a general linear supergroup G = GL(m|n), we consider a natural isomorphism φ : G → U − × Gev × U + , where Gev 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], extending results established in [8] and [7].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 ∈ Γ.

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“…For a general linear supergroup G = GL(m|n), we consider a natural isomorphism φ : G → U − × Gev × U + , where Gev 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], extending results established in [8] and [7].…”