Free Product Formulae for Quantum Permutation Groups

Associated to a finite graph X is its quantum automorphism group G(X). We prove a formula of type G(X ∗ Y ) = G(X) ∗w G(Y ), where ∗w is a free wreath product. Then we discuss representation theory of free wreath products, with the conjectural formula μ(G ∗w H) = 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. Introduction A quantum group is an abstract object, dual to a Hopf algebra. Finite quantum groups are those which are dual to finite dimensional Hopf algebras. A surprising fact, first noticed by Wang in [24], is that the quantum group corresponding to the Hopf algebra C(Z2 ∗ Z2) has a faithful action on the set {1, 2, 3, 4}. This quantum group, which is of course not finite, is a so-called quantum permutation group. In general, a quantum permutation group G is described by a special type of Hopf C-algebra A, according to the heuristic formula A = C(G). See [2], [24]. The simplest case is when A is commutative. Here G is a subgroup of the symmetric group Sn. This situation is studied by using finite group techniques. In general A is not commutative, and infinite dimensional. In this case G is a non-classical, non-finite compact quantum group. There is no analogue of a Lie algebra in this situation, but several representation theory methods, due to Woronowicz, are available ([25], [26]). A useful point of view comes from the heuristic formula A = C(Γ). Here Γ is a discrete quantum group, obtained as a kind of Pontrjagin dual of G. Number of discrete group techniques are known to apply to this situation. See e.g. [18], [19]. It is also known from [2] that quantum permutation groups are in one-to-one correspondence with subalgebras of spin planar algebras constructed in [11], [12]. Summarizing, a quantum permutation group should be regarded as a mixture of finite, compact and discrete groups, with a flavor of statistical mechanics, knot invariants and planar algebras. Several results are obtained along these lines in [1], [2], [3], [4], [5]. The aim of this work is to bring into the picture some free probability techniques. The starting point is the classical formula G(X . . .X) = G(X)×wG(Xn) for usual symmetry groups. Here X is a finite connected graph, Xn is a set having n elements, X . . .X is the disjoint union of n copies of X, and ×w is a wreath product. A series of free quantum analogues and 2000 Mathematics Subject Classification. 16W30 (46L37, 46L54).