Essential dimension of algebraic groups, including bad characteristic

TL;DR

This paper establishes upper bounds on the essential dimension of simple algebraic groups over algebraically closed fields across all characteristics, improving understanding of their complexity and representation theory.

Contribution

It provides new upper bounds on the essential dimension for most simple algebraic groups, including in bad characteristic, and computes generic stabilizers for adjoint groups.

Findings

Bounds are at most dim(G) - 2(rank(G)) - 1 for most groups.

Essential dimension of spin and half-spin groups grows exponentially with rank.

Bounds are comparable or better than characteristic zero cases.

Abstract

We give upper bounds on the essential dimension of (quasi-)simple algebraic groups over an algebraically closed field that hold in all characteristics. The results depend on showing that certain representations are generically free. In particular, aside from the cases of spin and half-spin groups, we prove that the essential dimension of a simple algebraic group $G$ of rank at least two is at most $\mathrm{dim}(G) - 2(\mathrm{rank}(G)) - 1$. It is known that the essential dimension of spin and half-spin groups grows exponentially in the rank. In most cases, our bounds are as good or better than those known in characteristic zero and the proofs are shorter. We also compute the generic stabilizer of an adjoint group on its Lie algebra.