Essential dimension of reductive groups via generically free representations

TL;DR

This paper introduces a straightforward method to compute the essential dimension of split reductive groups using generically free representations, enabling exact calculations for certain classical and exceptional groups.

Contribution

It provides a new simple approach to determine upper bounds on essential dimension, leading to exact values for various reductive groups, extending previous results.

Findings

Exact essential dimension values for classical types and E6 groups.

Extended previous work to all classical types and E6.

Unified approach for different reductive group types.

Abstract

We provide a simple method to compute upper bounds on the essential dimension of split reductive groups with finite or connected center by means of their generically free representations. Combining our upper bound with previously known lower bound, the exact value of the essential dimension is calculated for some types of reductive groups. As an application, we determine the essential dimension of a semisimple group of classical type or $E_{6}$, and its strict reductive envelope under certain conditions on its center. This extends previous works on simple simply connected groups of type $B$ or $D$ by Brosnan-Reichstein-Vistoli and Chernousov-Merkurjev, strict reductive envelopes of groups of type $A$ by Cernele-Reichstein, and semisimple groups of type $B$ by the authors to any classical type and type $E_{6}$ in a uniform way.