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free poisson law

Journal of the Institute of Mathematics of Jussieu (1)

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FREE PRODUCT FORMULAE FOR QUANTUM PERMUTATION GROUPS
Teodor Banica, Julien Bichon
Journal:

Journal of the Institute of Mathematics of Jussieu / Volume 6 / Issue 3 / July 2007

Published online by Cambridge University Press:

20 October 2006, pp. 381-414

Print publication:

July 2007
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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.