Asymptotic properties of tensor powers in symmetric tensor categories

Let G be a group and V a finite dimensional representation of G over an algebraically closed field k of characteristic p>0. Let d(n)(V) be the number of indecomposable summands of V circle times n of nonzero dimension mod p. It is easy to see that there exists a limit delta(V) := lim n ->infinity d(n)(V)1/n, which is positive (and >= 1) if f V has an indecomposable sum m and of nonzero dimension mod p. We show that in this case the number c(V) := lim in f(n)->infinity dn(V)delta(V)n is an element of[0,1] is strictly positive and log(c(V)(-1))=O(delta(V)(2)), and moreover this holds for any symmetric tensor category over k of moderate growth. Furthermore, we conjecture that in fact log(c(V)(-1))=O(delta(V)) (which would be sharp), and prove this for p=2,3; in particular, for p= 2 we show that c(V)>= 3-43(delta)(V)+1. The proofs are based on the characteristic p version of Deligne's theorem for symmetric tensor categories obtained in [CEO]. We also conjecture a classification of semisimple symmetric tensor categories of moderate growth which is interesting in its own right and implies the above conjecture for all p, and illustrate this conjecture by describing the semi simplification of the modular representation category of a cyclic p-group. Finally, we study the asymptotic behavior of the de-composition ofV circle times nin characteristic zero using Deligne's theorem and the Macdonald-Mehta-Opdam identity.