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Free product formulae for quantum permutation groups

Teodor Banica·Julien Bichon
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Abstract

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

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Teodor Banica, Julien Bichon. 2006-03-01. Free product formulae for quantum permutation groups. https://arxiv.org/abs/math/0503461

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