Lower bounds on the essential dimension of reductive groups

Danny Ofek

Lower bounds on the essential dimension of reductive groups

Group Theory 2025-10-27 v3 Algebraic Geometry

Authors: Danny Ofek

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.

Keywords

upper bounds bound combinatorial group theory

Cite

@article{arxiv.2411.03789,
  title  = {Lower bounds on the essential dimension of reductive groups},
  author = {Danny Ofek},
  journal= {arXiv preprint arXiv:2411.03789},
  year   = {2025}
}

Comments

52 pages; Added a french translation of the abstract, corrected a few typos and made small improvements to the exposition; To appear in Annales scientifiques de l'Ecole normale sup\'erieure

Lower bounds on the essential dimension of reductive groups

Abstract

Keywords

Cite

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