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UID:8398@i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20140414T110000
DTEND;TZID=Europe/Paris:20140414T120000
DTSTAMP:20241120T210400Z
URL:https://www.i2m.univ-amu.fr/evenements/expanseurs-quantiques-et-croiss
 ance-des-representations-de-groupes/
SUMMARY:Gilles Pisier (IMJ-PRG\, Sorbonne Université\, Paris): Expanseurs 
 quantiques et croissance des représentations de groupes
DESCRIPTION:Gilles Pisier: Quantum expanders and growth of group representa
 tions\nLet $\\pi$ be a finite dimensional unitary representation of a grou
 p $G$ with a generating symmetric $n$-element set $S\\subset G$. Fix $\\vp
 &gt\;0$. Assume that the spectrum of $|S|^{-1}\\sum_{s\\in S} \\pi(s) \\ot
 imes \\overline{\\pi(s)}$ is included in $ [-1\, 1-\\vp]$ (so there is a s
 pectral gap $\\ge \\vp$). Let $r'_N(\\pi)$ be the number of distinct irred
 ucible representations of dimension $\\le N$ that appear in $\\pi$. Then l
 et $R_{n\,\\vp}'(N)=\\sup r'_N(\\pi)$ where the supremum runs over all $\\
 pi$ with ${n\,\\vp}$ fixed. We prove that there are positive constants $\\
 delta_\\vp$ and $c_\\vp$ such that\, for all sufficiently large integer $n
 $ (i.e. $n\\ge n_0$ with $n_0$ depending on $\\vp$) and for all $N\\ge 1$\
 , we have $\\exp{\\delta_\\vp nN^2} \\le R'_{n\,\\vp}(N)\\le \\exp{c_\\vp 
 nN^2}$. The same bounds hold if\, in $r'_N(\\pi)$\, we count only the numb
 er of distinct irreducible representations of dimension exactly $= N$.\nht
 tps://arxiv.org/abs/1503.07937\n\n
ATTACH;FMTTYPE=image/jpeg:https://www.i2m.univ-amu.fr/wp-content/uploads/2
 020/01/Gilles_Pisier.jpg
CATEGORIES:Séminaire,Analyse et Géométrie
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DTSTART:20140330T030000
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