Deligne categories and the limit of categories $Rep(GL(m|n))$

Abstract

For each integer $t$ a tensor category $V_t$ is constructed, such that exact tensor functors $V_t⟶C$ classify dualizable $t$-dimensional objects in $C$ not annihilated by any Schur functor. This means that $V_t$ is the "abelian envelope" of the Deligne category $Rep(G{L}_t)$. Any tensor functor $Rep(G{L}_t)⟶C$ is proved to factor either through $V_t$ or through one of the classical categories $Rep(GL(m|n))$ with $m-n=t$. The universal property of $V_t$ implies that it is equivalent to the categories $Re{p}_{Rep(G{L}_{t_1})\otimes Rep(G{L}_{t_2})}(GL(X),\epsilon )$, ($t=t_1+t_2$, $t_1$ not integer) suggested by Deligne as candidates for the role of abelian envelope.

v3: lemma added to section 9, v4: some typos fixed, v5: fixed support acknowledgement, v6: minor fix in definition of abelian envelope, v7: fixed typo in preliminaries

References in corpus (1)

• Stability patterns in representation theory

Cited by in corpus (5)

• Semi-infinite highest weight categories
• Monoidal abelian envelopes
• Jacobson-Morozov Lemma for Algebraic Supergroups
• The indecomposable objects in the center of Deligne's category $Rep(S_t)$
• Monoidal abelian envelopes and a conjecture of Benson--Etingof