Counting fine gradings on matrix algebras and on classical simple Lie algebras

Abstract: Known classification results allow us to find the number of (equivalence classes of) fine gradings on matrix algebras and on classical simple Lie algebras over an algebraically closed field $\mathbb{F}$ (assuming $\mathrm{char} \mathbb{F}\ne 2$ in the Lie case). The computation is easy for matrix algebras and especially for simple Lie algebras of type $B_r$ (the answer is just $r+1$), but involves counting orbits of certain finite groups in the case of Series $A$, $C$ and $D$. For $X\in{A,C,D}$, we determine the exact number of fine gradings, $N_X(r)$, on the simple Lie algebras of type $X_r$ with $r\le 100$ as well as the asymptotic behaviour of the average, $\hat N_X(r)$, for large $r$. In particular, we prove that there exist positive constants $b$ and $c$ such that $\exp(br^{2/3})\le\hat N_X(r)\le\exp(cr^{2/3})$. The analogous average for matrix algebras $M_n(\mathbb{F})$ is proved to be $a\ln n+O(1)$ where $a$ is an explicit constant depending on $\mathrm{char} \mathbb{F}$.