A combinatorial approach to Donkin-Koppinen filtrations of general linear supergroups. Issue 7 (18th June 2021) - Marko, František
Abstract: For a general linear supergroup G = GL ( m | n ), we consider a natural isomorphism ϕ : G → U − × G e v × 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 i j ) 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 G L ( 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 . We apply combinatorial techniques, using generalized bideterminants, to determine a basis of G -superbimodules appearing in Donkin-Koppinen filtration of O Γ ( K [ G ] ), considered initially in [9 ].