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

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 $C{k}^{-{\alpha }_p}(\text{dim}T{)}^k<b_k^{T,{\text{SL}}_2(K)}<D{k}^{-{\alpha }_p}(\text{dim}T{)}^k$ for some $C,D>0$ where ${\alpha }_p=1-(1/2){\log }_p(\frac{p+1}{2}).$