Free Product Formulae for Quantum Permutation Groups

Abstract

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) * 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.

Graphs having no quantum symmetry

“…These quantum groups are studied in [9], [10] and [4], [3], then in [5], [6]. The motivation comes from certain combinatorial aspects of subfactors, free probability, and statistical mechanical models.…”

“…An explanation regarding C 4 is proposed in [5]: this graph is exceptional in the series because it is the one having non-trivial disconnected complement. Indeed, the quantum symmetry group is the same for a graph and for its complement, and duplication of graphs corresponds to free wreath products, known from [10] to be highly non-commutative operations.…”

Quantum automorphism groups of vertex-transitive graphs of order ≤ 11

“…This enhances previous classification work from [2][3][4], where we have n ≤ 9, also with one graph omitted.…”

“…This was started independently by the authors in [2,3] and [7,8], and continued in the joint paper [4]. The notion that emerges from this work is that of quantum automorphism group of a vertex-transitive graph.…”

Hopf algebras having a dense big cell

“…However there are other very interesting examples, introduced e.g. in the papers [18,33,32,35,8], related to free products and free probability (see for example [4,5,6,26,7] for these kinds of developments), that do not fit into the q-deformation scheme.…”