Asymptotic properties of tensor powers in symmetric tensor categories

Abstract

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 $V^{\otimes n}$ of nonzero dimension mod p. It is easy to see that there exists a limit $\delta (V):={\lim }_{n\to \infty }{d}_n(V{)}^{1/n}$, which is positive (and $\ge 1$) iff V has an indecomposable summand of nonzero dimension mod p. We show that in this case the number $$c(V):=\underset{n\to \infty }{\mathrm{liminf}}\frac{d_n(V)}{\delta (V{)}^n}\in [0,1]$$ 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 $c(V)\ge 3^{-\frac{4}{3}\delta (V)+1}$. The proofs are based on the characteristic p version of Deligne's theorem for symmetric tensor categories obtained in earlier work of the authors. 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 semisimplification of the modular representation category of a cyclic p-group. Finally, we study the asymptotic behavior of the decomposition of $V^{\otimes n}$ in characteristic zero using Deligne's theorem and the Macdonald-Mehta-Opdam identity.