Quantum expanders and growth of group representations Academic Article uri icon

abstract

  • Let $pi$ be a finite dimensional unitary representation of a group $G$ with a generating symmetric $n$-element set $Ssubset G$. Fix $vp>0$. Assume that the spectrum of $|S|^{-1}sum_{sin S} pi(s) otimes overline{pi(s)}$ is included in $ [-1, 1-vp]$ (so there is a spectral gap $ge vp$). Let $r'_N(pi)$ be the number of distinct irreducible representations of dimension $le N$ that appear in $pi$. Then let $R_{n,vp}'(N)=sup r'_N(pi)$ 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. $nge n_0$ with $n_0$ depending on $vp$) and for all $Nge 1$, 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 number of distinct irreducible representations of dimension exactly $= N$.

published proceedings

  • Annales de la facult des sciences de Toulouse Mathmatiques

author list (cited authors)

  • Pisier, G.

citation count

  • 0

complete list of authors

  • Pisier, Gilles

publication date

  • March 2015