On a conjecture of Lionel Schwartz about the eigenvalues of Lannes' T-functor

Nguyen Dang Ho Hai

doi:10.1016/j.crma.2014.12.006

Algebra/Topology

On a conjecture of Lionel Schwartz about the eigenvalues of Lannes' T-functor
[À propos d'une conjecture de Lionel Schwartz sur les valeurs propres du foncteur T de Lannes]

Présenté par : the Editorial Board

Nguyen Dang Ho Hai ¹

¹ University of Hue, College of Sciences, 77 Nguyen Hue Street, Hue City, Viet Nam

Comptes Rendus. Mathématique, Volume 353 (2015) no. 3, pp. 197-202.

Résumé
Abstract

Étant donné un nombre premier p, on note $K^{red} (U)$ le groupe de Grothendieck engendré par les classes d'isomorphisme de modules réduits injectifs indécompsables de la catégorie des modules instable sur l'algèbre de Steenrod modulo p. On note $K_{n}^{red} (U)$ , $n \in N$ , le sous-groupe de $K^{red} (U)$ engendré par les facteurs indécomposables de $H^{⁎} B {(Z / p)}^{n}$ . On décrit dans cette note une stratégie pour démontrer la conjecture suivante due à Lionel Schwartz : l'opérateur induit par le foncteur T de Lannes sur l'espace vectoriel rationnel $Q \otimes_{Z} K_{n}^{red} (U)$ est diagonalisable et a pour valeurs propres $1, p, \dots, p^{n - 1}, p^{n}$ de multiplicités $p^{n} - p^{n - 1}, p^{n - 1} - p^{n - 2}, \dots, p - 1, 1$ , respectivement.

Given a prime p, let $K^{red} (U)$ denote the Grothendieck group generated by the isomorphism classes of indecomposable injective reduced modules in the category of unstable modules over the mod p Steenrod algebra. Let $K_{n}^{red} (U)$ , $n \in N$ , denote the subgroup of $K^{red} (U)$ generated by the indecomposable summands of $H^{⁎} B {(Z / p)}^{n}$ . We describe in this note a strategy for the proof of the following conjecture of Lionel Schwartz: the operator induced by Lannes' T-functor on the rational vector space $Q \otimes_{Z} K_{n}^{red} (U)$ is diagonalizable and has eigenvalues $1, p, \dots, p^{n - 1}, p^{n}$ with multiplicities $p^{n} - p^{n - 1}, p^{n - 1} - p^{n - 2}, \dots, p - 1, 1$ , respectively.

Reçu le : 2014-10-27
Accepté le : 2014-12-19
Publié le : 2015-01-28

DOI : 10.1016/j.crma.2014.12.006

Affiliations des auteurs :

Nguyen Dang Ho Hai ¹

¹ University of Hue, College of Sciences, 77 Nguyen Hue Street, Hue City, Viet Nam

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     author = {Nguyen Dang Ho Hai},
     title = {On a conjecture of {Lionel} {Schwartz} about the eigenvalues of {Lannes'} {T-functor}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {197--202},
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     doi = {10.1016/j.crma.2014.12.006},
     language = {en},
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Nguyen Dang Ho Hai. On a conjecture of Lionel Schwartz about the eigenvalues of Lannes' T-functor. Comptes Rendus. Mathématique, Volume 353 (2015) no. 3, pp. 197-202. doi : 10.1016/j.crma.2014.12.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.12.006/

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Cité par Sources :

^☆ This work was initiated while the author was a CNRS researcher at LAREMA, Angers. The author would like to thank the CNRS for financial support, LIAFV for travel support and LAREMA for a peaceful working environment. It is a pleasure for the author to thank Geoffrey Powell and Jean Lannes for valuable discussions on the Singer functor and the Segal conjecture, and Lionel Schwartz for his special interest in this work. He also would like to thank the referee for helpful comments that greatly improved the manuscript. The author is partially supported by the NAFOSTED project “Algebraic Topology and Representation Theory”.

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