Integral objects and Deligne's category Rep(S_t)

TL;DR

This paper investigates Deligne's category Rep(S_t) for non-integer t, providing negative answers to questions about abelian semisimple tensor categories and demonstrating that Rep(S_t) is not Schur-finite, thus not equivalent to supergroup representations.

Contribution

It offers new insights into Deligne's categories, showing they are not Schur-finite and addressing questions on abelian semisimple tensor categories.

Findings

Rep(S_t) is not Schur-finite

Rep(S_t) is not tensor equivalent to supergroup representations

Provides negative answers to certain categorical questions

Abstract

We give negative answers to certain questions on abelian semisimple tensor categories raised by Bruno Kahn and Charles A. Weibel in connection with the preprint of Kahn "On the multiplicities of a motive" (arXiv:math/0610446), now published as Parts I and IV of "Zeta functions and motives", Pure Appl. Math. Q. 5 (2009), no. 1, part 2, 507-570. For the most interesting examples we used Deligne's category Rep(S_t,F) of representations of the "symmetric group S_t with t not an integer" with F any algebraically closed field of characteristic zero. This is an interesting family of tensor categories, "new" in some sense, interpolating the representations of the symmetric groups. Among other things we give two proofs that this category is not Schur-finite, showing hence explicitly that it is not tensor equivalent to a category of superepresentations of a supergroup.