Fractal behavior of tensor powers of tilting modules of $\text{SL}_2$

TL;DR

This paper investigates the asymptotic behavior of the number of indecomposable summands in tensor powers of tilting modules for SL_2 over algebraically closed fields, revealing a fractal-like growth pattern influenced by the characteristic p.

Contribution

It establishes precise bounds on the growth of indecomposable summands in tensor powers of tilting modules for SL_2, highlighting a fractal behavior depending on the characteristic p.

Findings

Bounds on the number of indecomposable summands grow exponentially with tensor power.

The growth rate involves a characteristic-dependent exponent lpha_p.

The behavior exhibits a fractal-like pattern influenced by the prime characteristic.

Abstract

Given a group $G$ and $V$ a representation of $G$, denote the number of indecomposable summands of $V^{\otimes k}$ by $b_k^{G, V}$. Given a tilting representation $T$ of $\text{SL}_2(K)$ where $K=\overline{K}$ and of characteristic $p>2$, we show that $Ck^{-\alpha_p}(\text{dim} T)^k<b_k^{T, \text{SL}_2(K)}<Dk^{-\alpha_p}(\text{dim} T)^k$ for some $C, D>0$ where $\alpha_p=1-(1/2)\log_p(\frac{p+1}{2}).$