## Data

• Title: Fractal behavior of tensor powers of the two dimensional space in prime characteristic
• Authors: Kevin Coulembier, Pavel Etingof, Victor Ostrik and Daniel Tubbenhauer
• Status: preprint. Last update: Mon, 27 May 2024 03:17:22 UTC
For a group (scheme) $$G$$ and a finite dimensional (rational) $$G$$-representation $$V$$ let $$b_n=b_n(V)$$ denote the number of indecomposable summands of $$V^{\otimes n}$$, and $$l_n=l_n(V)$$ the length of $$V^{\otimes n}$$. We make the Ansatz that: we have $b_n\sim h(n)\cdot n^{a}\cdot(\dim V)^{n},\quad\mbox{and}\quad l_n\sim h'(n)\cdot n^{a'}\cdot(\dim V)^{n}$ for bounded functions $$h$$ and $$h'$$ and $$a,a'\in\mathbb{R}$$.
For example, take the vector representation $$V=\mathbb{C}^2$$ of $$\mathrm{SL}_{2}(\mathbb{C})$$, and let $b_n$ be the length of (equivalently the number of indecomposable summands in) $$V^{\otimes n}$$. One sees that $$C\cdot n^{-1}\cdot 2^n\leq b_n\leq 2^n,C\in\mathbb{R}_{>0}$$, since the largest possible summand in $$V^{\otimes n}$$ has dimension $$n+1$$, while $$\dim V^{\otimes n}=2^n$$. In fact, $$b_n\sim a_n=D\cdot n^{-1/2}\cdot 2^n$$, where $$D=\sqrt{2/\pi}$$. The final, and most difficult, step is to see that the variance is $$|b_n-a_n|\sim E\cdot n^{-3/2}\cdot 2^n,E\in\mathbb{R}_{>0}$$. The plots are:
This paper addresses the same type of questions for the prime characteristic version of the above. Specifically, for the algebraic group $$G=\mathrm{SL}_{2}(\bar{\mathbb{F}}_p)$$ and its vector representation $$\bar{\mathbb{F}}_p^2$$ we proved that the Ansatz above is valid and computed $$a=-1+\tfrac{1}{2}\log_{p}\tfrac{p+1}{2}$$ and $$a'=\log_p(2p-1)$$.