Sharp bounds for frame counts and setwise stabilizers in classical groups
Let $k$ be a field, let $V=k^n$, and let $G$ be a classical group acting on $V$. For a finite subset $E\subset V$ and a basis $u=(u_1,\dots,u_n)$ of $V$, we study the set of $G$-frames of type $u$ contained in $E$, or, equivalently, the set $T_{E,u}^G:=\{g\in G(k)\ |\ g u_i\in E\text{ for each $i$}\}$. In the cases described below, we prove estimates of the form $|T_{E,u}^G|\ll_n |E|^\alpha$ that are uniform over all fields, with sharp exponents in this uniform setting. For $G=\operatorname{SL}_n$, the uniform sharp exponent is $n-1/n$. For orthogonal groups in dimensions $n=2,3$, with $\operatorname{char}(k)\neq 2$, the uniform sharp exponent is $n/2$. We propose an algebro-geometric Brascamp--Lieb inequality which would lead to the orthogonal exponent $n/2$ in all dimensions. For the related setwise stabilizer $R_E^G$ of $E$ in $G(k)$, where $E\subset V$ is finite and spans $V$, we also prove the sharp characteristic-zero bound $|R_E^G|\ll_n E|^{\operatorname{rank}G}$ for the special linear, orthogonal, and symplectic groups, where $\operatorname{rank}G$ denotes the absolute rank.