Asymptotic properties of tensor powers in symmetric tensor categories

arXiv2301.09804MaRDI QIDQ6508535

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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 $V^{otimesn}$ of nonzero dimension mod p. It is easy to see that there exists a limit $delta(V):=li{m}_{noinfty}{d}_n(V{)}^{1/n}$, which is positive (and $ge1$) iff V has an indecomposable summand of nonzero dimension mod p. We show that in this case the number ${\displaystyle }$ c(V):=liminf_{n o infty} frac{d_n(V)}{delta(V)^n}in [0,1] ${\displaystyle }$ is strictly positive and ${\displaystyle }$ log (c(V)^{-1})=O(delta(V)^2), ${\displaystyle }$ and moreover this holds for any symmetric tensor category over k of moderate growth. Furthermore, we conjecture that in fact ${\displaystyle }$ log(c(V)^{-1})=O(delta(V)) ${\displaystyle }$ (which would be sharp), and prove this for p=2,3; in particular, for p=2 we show that $c(V)ge{3}^{-frac43delta(V)+1}$. 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 $V^{otimesn}$ in characteristic zero using Deligne's theorem and the Macdonald-Mehta-Opdam identity.

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