Asymptotic properties of tensor powers in symmetric tensor categories

TL;DR

This paper investigates the asymptotic behavior of indecomposable summands in tensor powers within symmetric tensor categories over fields of positive characteristic, establishing bounds and conjectures on their growth rates.

Contribution

It introduces new bounds on the growth rate of indecomposable summands in tensor powers and proposes conjectures on their asymptotic behavior, extending Deligne's theorem to positive characteristic.

Findings

Established that $c(V)$ is strictly positive when $ ext{delta}(V)>1$

Proved bounds on $ ext{log}(c(V)^{-1})$ in terms of $ ext{delta}(V)$ for p=2,3

Conjectured sharper bounds and classified semisimple categories of moderate growth

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 $$ c(V):=\liminf_{n\to \infty} \frac{d_n(V)}{\delta(V)^n}\in [0,1] $$ 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.