Lower bounds on the essential dimension of reductive groups

TL;DR

This paper develops a new technique to establish lower bounds on the essential dimension of split reductive groups, improving known bounds for several groups including E8 and PGL_n, using decompositions of loop torsors.

Contribution

It introduces a novel method based on loop torsor decompositions to prove lower bounds, strengthening results for various split simple algebraic groups.

Findings

Strengthened lower bounds for E8 and other groups

Simplified proof of Merkurjev's lower bound for PGL_n

New technique based on loop torsor decompositions

Abstract

We introduce a technique for proving lower bounds on the essential dimension of split reductive groups. As an application, we strengthen the best previously known lower bounds for various split simple algebraic groups, most notably for the exceptional group $E_8$. In the case of the projective linear group $\operatorname{PGL}_n$, we recover A. Merkurjev's celebrated lower bound with a simplified proof. Our technique relies on decompositions of loop torsors over valued fields due to P. Gille and A. Pianzola.