The dimension of solution sets to systems of equations in algebraic groups

TL;DR

This paper extends the Gordon--Rodriguez-Villegas theorem from finite groups to algebraic groups, establishing lower bounds on the dimension of solution sets for systems of equations in algebraic groups.

Contribution

It provides new analogues of divisibility and dimension results for solution sets in algebraic groups, generalizing known finite group theorems.

Findings

Dimension of irreducible components is at least the dimension of G.

Solutions to systems of equations have structured dimension properties.

Results imply lower bounds on solution set dimensions in algebraic groups.

Abstract

The Gordon--Rodriguez-Villegas theorem says that, in a finite group, the number of solutions to a system of coefficient-free equations is divisible by the order of the group if the rank of the matrix composed of the exponent sums of $j$-th unknown in $i$-th equation is less than the number unknowns. We obtain analogues of this and similar facts for algebraic groups. In particular, our results imply that the dimension of each irreducible component of the variety of homomorphisms from a finitely generated group with infinite abelianisation into an algebraic group $G$ is at least $\dim G$.