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*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)μ (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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