On classical tensor categories attached to the irreducible representations of the general linear supergroups GL(n|n) - Heidersdorf, Thorsten; Weissauer, Rainer (Selecta mathematica, 2023)
We study the quotient of $$\mathcal {T}_n = Rep(GL(n|n))$$by the tensor ideal of negligible morphisms. If we consider the full subcategory $$\mathcal {T}_n^+$$of $$\mathcal {T}_n$$of indecomposable summands in iterated tensor products of irreducible representations up to parity shifts, its quotient is a semisimple tannakian category $$Rep(H_n)$$where $$H_n$$is a pro-reductive algebraic group. We determine the $$H_n$$and the groups $$H_{λ }$$corresponding to the tannakian subcategory in $$Rep(H_n)$$generated by an irreducible representation $$L(λ )$$. This gives structural information about the tensor category Rep(GL(n|n)), including the decomposition law of a tensor product of irreducible representations up to summands of superdimension zero. Some results are conditional on a hypothesis on 2-torsion in $$π _0(H_n)$$.
Gesehen am 31.07.2023