Integral objects and Deligne's category Rep(S_t)

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.