Asymptotic properties of tensor powers in symmetric tensor categories
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866929246770823168 |
|---|---|
| author | Coulembier, Kevin Etingof, Pavel Ostrik, Victor |
| author_facet | Coulembier, Kevin Etingof, Pavel Ostrik, Victor |
| contents | 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 $δ(V):=\lim_{n\to \infty}d_n(V)^{1/n}$, 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 $$ c(V):=\liminf_{n\to \infty} \frac{d_n(V)}{δ(V)^n}\in [0,1] $$ is strictly positive and $$ \log (c(V)^{-1})=O(δ(V)^2), $$ and moreover this holds for any symmetric tensor category over k of moderate growth. Furthermore, we conjecture that in fact $$ \log(c(V)^{-1})=O(δ(V)) $$ (which would be sharp), and prove this for p=2,3; in particular, for p=2 we show that $c(V)\ge 3^{-\frac{4}{3}δ(V)+1}$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_09804 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Asymptotic properties of tensor powers in symmetric tensor categories Coulembier, Kevin Etingof, Pavel Ostrik, Victor Representation Theory Category Theory 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 $δ(V):=\lim_{n\to \infty}d_n(V)^{1/n}$, 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 $$ c(V):=\liminf_{n\to \infty} \frac{d_n(V)}{δ(V)^n}\in [0,1] $$ is strictly positive and $$ \log (c(V)^{-1})=O(δ(V)^2), $$ and moreover this holds for any symmetric tensor category over k of moderate growth. Furthermore, we conjecture that in fact $$ \log(c(V)^{-1})=O(δ(V)) $$ (which would be sharp), and prove this for p=2,3; in particular, for p=2 we show that $c(V)\ge 3^{-\frac{4}{3}δ(V)+1}$. 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. |
| title | Asymptotic properties of tensor powers in symmetric tensor categories |
| topic | Representation Theory Category Theory |
| url | https://arxiv.org/abs/2301.09804 |