Invariant derivations and trace bounds

Skip Garibaldi; Robert M. Guralnick; Eric M. Rains

doi:10.5802/jep.310

Invariant derivations and trace bounds
[Dérivations invariantes et bornes sur la trace]

Skip Garibaldi ¹ ; Robert M. Guralnick ² ; Eric M. Rains ³

¹ IDA Center for Communications Research-La Jolla, San Diego, CA 92121, USA
² Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, USA
³ Department of Mathematics, California Institute of Technology, 1200 E California Blvd, Pasadena, CA 91125, USA

Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 1229-1287.

Abstract (VO)
Résumé

About 20 years ago, J-P. Serre announced a bound on the trace of elements of compact Lie groups under the adjoint representation together with related results, provided indications of his proofs, and invited a better proof. This note provides a new, general method for proving such bounds; uses that method to derive Serre’s bounds; gives a second proof of Serre’s announced results that (we learned) closely follows his original argument; and provides lower bounds for traces of other representations of compact Lie groups and for Brauer characters of finite groups.

Il y a une vingtaine d’années, J-P. Serre annonçait une borne sur la trace des éléments des groupes de Lie compacts sous la représentation adjointe, ainsi que des résultats correspondants, fournissait des indications sur ses preuves et invitait à une meilleure démonstration. Cette note propose une nouvelle méthode générale pour prouver ces bornes ; utilise cette méthode pour dériver les bornes de Serre ; donne une seconde démonstration des résultats annoncés par Serre qui (nous l’avons appris) suit étroitement son argument initial ; et fournit des bornes inférieures pour les traces d’autres représentations de groupes de Lie compacts et pour les caractères de Brauer des groupes finis.

Reçu le : 2024-02-15
Accepté le : 2025-08-13
Publié le : 2025-08-25

DOI : 10.5802/jep.310

Classification : 22E47, 22C05, 22G20
Keywords: Invariant derivation, compact group, trace, Brauer character, adjoint
Mots-clés : Dérivation invariante, groupe compact, trace, caractère de Brauer, adjoint

Affiliations des auteurs :

Skip Garibaldi ¹ ; Robert M. Guralnick ² ; Eric M. Rains ³

¹ IDA Center for Communications Research-La Jolla, San Diego, CA 92121, USA
² Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, USA
³ Department of Mathematics, California Institute of Technology, 1200 E California Blvd, Pasadena, CA 91125, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Invariant derivations and trace bounds},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {1229--1287},
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Skip Garibaldi; Robert M. Guralnick; Eric M. Rains. Invariant derivations and trace bounds. Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 1229-1287. doi : 10.5802/jep.310. https://jep.centre-mersenne.org/articles/10.5802/jep.310/

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