Quantum expanders and growth of group representations

Gilles Pisier

doi:10.5802/afst.1541

Gilles Pisier ¹

¹ Texas A&M University, College Station, TX 77843, USA and Université Paris VI, Inst. Math. Jussieu, Équipe d’Analyse Fonctionnelle, Case 186, 75252 Paris Cedex 05, France

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 2, pp. 451-462.

Résumé
Abstract

Soit $π$ une représentation unitaire de dimension finie d’un groupe $G$ munie d’un ensemble générateur symétrique $S \subset G$ à $n$ -éléments. Fixons $ε > 0$ et supposons que le spectre de ${| S |}^{- 1} \sum_{s \in S} π (s) \otimes \bar{π (s)}$ est inclus dans $[- 1, 1 - ε]$ (il y a donc un trou spectral $\geq ε$ ). Soit $r_{N}^{'} (π)$ le nombre de représentations irréductibles distinctes de dimension $\leq N$ qui apparaissent dans la décomposition de $π$ . Soit alors $R_{n, ε}^{'} (N) = sup r_{N}^{'} (π)$ où le sup court sur toutes les $π$ possibles avec $n, ε$ fixés. Nous démontrons l’existence de constantes positives $δ_{ε}$ et $c_{ε}$ telles que, pour tout entier $n$ suffisamment grand (i.e. $n \geq n_{0}$ ou $n_{0}$ peut dépendre de $ε$ ) et pour tout $N \geq 1$ , on a $exp δ_{ε} n N^{2} \leq R_{n, ε}^{'} (N) \leq exp c_{ε} n N^{2}$ . Les mêmes bornes sont valables si, dans $r_{N}^{'} (π)$ , on compte seulement le nombre de représentations irréductibles distinctes de dimension exactement $= N$ .

Let $π$ be a finite dimensional unitary representation of a group $G$ with a generating symmetric $n$ -element set $S \subset G$ . Fix $ε > 0$ . Assume that the spectrum of ${| S |}^{- 1} \sum_{s \in S} π (s) \otimes \bar{π (s)}$ is included in $[- 1, 1 - ε]$ (so there is a spectral gap $\geq ε$ ). Let $r_{N}^{'} (π)$ be the number of distinct irreducible representations of dimension $\leq N$ that appear in $π$ . Then let $R_{n, ε}^{'} (N) = sup r_{N}^{'} (π)$ where the supremum runs over all $π$ with $n, ε$ fixed. We prove that there are positive constants $δ_{ε}$ and $c_{ε}$ such that, for all sufficiently large integer $n$ (i.e. $n \geq n_{0}$ with $n_{0}$ depending on $ε$ ) and for all $N \geq 1$ , we have $exp δ_{ε} n N^{2} \leq R_{n, ε}^{'} (N) \leq exp c_{ε} n N^{2}$ . The same bounds hold if, in $r_{N}^{'} (π)$ , we count only the number of distinct irreducible representations of dimension exactly $= N$ .

Publié le : 2017-04-12

DOI : 10.5802/afst.1541

Affiliations des auteurs :

Gilles Pisier ¹

¹ Texas A&M University, College Station, TX 77843, USA and Université Paris VI, Inst. Math. Jussieu, Équipe d’Analyse Fonctionnelle, Case 186, 75252 Paris Cedex 05, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Gilles Pisier},
     title = {Quantum expanders and growth of group representations},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {451--462},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
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Gilles Pisier. Quantum expanders and growth of group representations. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 2, pp. 451-462. doi : 10.5802/afst.1541. https://afst.centre-mersenne.org/articles/10.5802/afst.1541/

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