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Quantum expanders and growth of group representations
Gilles Pisier
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 2, p. 451-462

Let π be a finite dimensional unitary representation of a group G with a generating symmetric n-element set SG. Fix ε>0. Assume that the spectrum of |S| -1 sS π(s)π(s) ¯ is included in [-1,1-ε] (so there is a spectral gap ε). Let r N ' (π) be the number of distinct irreducible representations of dimension N that appear in π. Then let R n,ε ' (N)=supr 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. nn 0 with n 0 depending on ε) and for all N1, we have expδ ε nN 2 R n,ε ' (N)expc ε nN 2 . The same bounds hold if, in r N ' (π), we count only the number of distinct irreducible representations of dimension exactly =N.

Soit π une représentation unitaire de dimension finie d’un groupe G munie d’un ensemble générateur symétrique SG à n-éléments. Fixons ε>0 et supposons que le spectre de |S| -1 sS π(s)π(s) ¯ est inclus dans [-1,1-ε] (il y a donc un trou spectral ε). Soit r N ' (π) le nombre de représentations irréductibles distinctes de dimension N qui apparaissent dans la décomposition de π. Soit alors R n,ε ' (N)=supr 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. nn 0 ou n 0 peut dépendre de ε) et pour tout N1, on a expδ ε nN 2 R n,ε ' (N)expc ε nN 2 . Les mêmes bornes sont valables si, dans r N ' (π), on compte seulement le nombre de représentations irréductibles distinctes de dimension exactement =N.

Published online : 2017-04-13
DOI : https://doi.org/10.5802/afst.1541
@article{AFST_2017_6_26_2_451_0,
     author = {Gilles Pisier},
     title = {Quantum expanders and growth of group representations},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 26},
     number = {2},
     year = {2017},
     pages = {451-462},
     doi = {10.5802/afst.1541},
     language = {en},
     url = {https://afst.centre-mersenne.org/item/AFST_2017_6_26_2_451_0}
}
Pisier, Gilles. Quantum expanders and growth of group representations. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 2, pp. 451-462. doi : 10.5802/afst.1541. afst.centre-mersenne.org/item/AFST_2017_6_26_2_451_0/

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