Free product formulae for quantum permutation groups

Abstract

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

31 pages