On Supergroups and their Semisimplified Representation Categories

Abstract

The representation category A = Rep(G, ǫ) of a supergroup scheme G has a largest proper tensor ideal, the ideal N of negligible morphisms. If we divide A by N we get the semisimple representation category of a pro-reductive supergroup scheme G red . We list some of its properties and determine G red in the case GL(m|1).

Mixed tensors of the general linear supergroup

“…Fully faithful follows from Schur's lemma in the semisimple tensor category T /N . Theorem 10.2 plays a crucial role in [Hei15]. An analogous theorem holds in the orthosymplectic case [CH17].…”

“…Theorem 10.2 plays a crucial role in [Hei15]. An analogous theorem holds in the orthosymplectic case [CH17].…”

“…Proof. The functor will factorize if ρ T is full [Hei15]. This follows from the commutative diagram since an indecomposable module maps to an irreducible module.…”

“…which is used in a crucial way in [Hei15]. Since we have character and dimension formulas for every mixed tensor by [CW12,Theorem 8.5.2], we get character and dimension formulas for Kostant modules and projective modules in section 8.1.…”

“…Then P ⊗ Z r (a) = ai P ⊗ L(a i ), P ⊗ Z r (a) = ai P ⊗ L(a i ) where the sums run over the composition factors L(a i ) of Z r (a) respectively Z r (a).All in all the only remaining unknown tensor products in the GL(m|1)-case are the tensor products Z r (a) ⊗ Z s (b), (r or s ≥ 2) and vice versa for the Anti-ZigZag-modules. If r, s are odd their tensor product decomposes as given by the Littlewood-Richardson Rule for GL(m−n) modulo contributions of superdimension 0[Hei15]. 8.3.…”