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 $p \gt 0$. Let $d_n (V)$ be the number of indecomposable summands of $V^{\oplus n}$ 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 $\geq 1$) $\operatorname{iff}$ $V$ has an indecomposable summand of nonzero dimension $\mod p$. We show that in this case the number\[c(V ) := \underset{n \to \infty}{\lim \inf} \frac{d_n(V)}{\delta(V)^n}$ \in [0, 1]\]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 $c(V) \geq 3^{\frac{4}{3} \delta (V)+1}$. The proofs are based on the characteristic $p$ version of Deligne’s theorem for symmetric tensor categories obtained in $\href{ https://dx.doi.org/10.4007/annals.2023.197.3.5}{[\textrm{CEO}}]$. 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 semisimplification of the modular representation category of a cyclic $p$-group. Finally, we study the asymptotic behavior of the decomposition of $V^{\oplus n}$ in characteristic zero using Deligne’s theorem and the Macdonald–Mehta–Opdam identity.