Asymptotics for number of indecomposable components of tensor powers of the natural $\mathrm{SL}_2$-module in odd characteristic

Abstract

Let $K$ be an algebraically closed field of odd characteristic $p$, let $G= {\rm SL}_2(K)$, and let $V$ be the natural representation of $G$. Let $b_k$ denote the number of $G$-indecomposable factors of $V^{\otimes k}$, counted with multiplicity, and let $\delta_p=1-\log_{p^2}\!\bigl(\tfrac{p+1}{2}\bigr)$. Then there exists a smooth, strictly positive, multiplicatively $p^2$-periodic function $\omega(t)$ such that $b_k$ is asymptotic to $\omega(k)k^{-\delta_p}2^k$. We also show that $t^{-\delta}\omega(t)$ arises as the limiting density of renormalized convolutions of rescaled copies of a positive weight $3/2$ theta function, obtained from the boundary heat flux of the Dirichlet heat kernel on $(0,p)$.