Growth Rates of the Number of Indecomposable Summands in Tensor Powers

Coulembier, Kevin; Ostrik, Victor; Tubbenhauer, Daniel

doi:10.1007/s10468-023-10245-7

Cited by 1 publication

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“…Asymptotic properties of tensor powers of a modular representation (or, more generally, an object of a symmetric tensor category) are an interesting and mysterious subject, about which rather little is known. It has recently been studied in the papers [B2,BS,CEO,EK,COT]. The goal of this paper is to continue this study.…”

Section: Growth Of the Non-negligible Part Of A Tensor Powermentioning

confidence: 98%

Asymptotic properties of tensor powers in symmetric tensor categories

Coulembier,

Etingof,

Ostrik

2024

Pure and Applied Mathematics Quarterly

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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 ⊗n of nonzero dimension mod p. It is easy to see that there exists a limit δ(V ) := lim n→∞ d n (V ) 1/n , which is positive (and ≥ 1) iff V has an indecomposable summand of nonzero dimension mod p. We show that in this case the numberand moreover this holds for any symmetric tensor category over k of moderate growth. Furthermore, we conjecture that in fact log(c(V ) −1 ) = O(δ(V )) (which would be sharp), and prove this for p = 2, 3; in particular, for p = 2 we show that c(V ) ≥ 3 − 4 3 δ(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 semisimplification of the modular representation category of a

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