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FREE PRODUCT FORMULAE FOR QUANTUM PERMUTATION GROUPS

Published online by Cambridge University Press:  20 October 2006

Teodor Banica  and
Julien Bichon
Teodor Banica
Affiliation:
Departement de Mathematiques, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France (banica@picard.ups-tlse.fr)
Julien Bichon
Affiliation:
Laboratoire de Mathematiques Appliquees, Université de Pau et des Pays de l’Adour, IPRA, Avenue de l’universite, 64000 Pau, France (bichon@univ-pau.fr)
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Abstract

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Associated to a finite graph $X$ is its quantum automorphism group $G(X)$. We prove a formula of type $G(X*Y)=G(X)*_{\mathrm{w}}G(Y)$, where $*_{\mathrm{w}}$ is a free wreath product. Then we discuss representation theory of free wreath products, with the conjectural formula $\mu(G*_{\mathrm{w}}H)=\mu(G)\boxtimes\mu(H)$, where $\mu$ is the associated spectral measure. This is verified in two situations: one using free probability techniques, the other one using planar algebras.

Keywords

free wreath productfree multiplicative convolutionfree Poisson lawPrimary 16W30Secondary 46L3746L54
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 6 , Issue 3 , July 2007 , pp. 381 - 414
Copyright
2006 Cambridge University Press