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By a free symmetric quantum group one means any element of the one-parameter sequence of compact matrix quantum groups defined by Wang in [Wang98] under the name quantum permutation groups. Each is a free counterpart of the symmetric group of the corresponding dimension .
Given , the free symmetric quantum group (or quantum permuation group on symbols) is the compact matrix quantum group where organizes the generators of the (unital) universal C*-algebra
where is the unit of the universal -algebra.
In other words, the entries of the fundamental corpresentation matrix of are projections, i.e., self-adjoint idempotents, and the entries of each row or column form a partition of unity, i.e., mutually orthogonal projections summing up (where the orthogonality is to mean and for all as can be shown). Those relations are commonly summarized by saying that is a magic unitary.
The fundamental corepresentation matrix of is in particular orthogonal. Hence, is a compact quantum subgroup of the free orthogonal quantum group .
If denotes the closed two-sided ideal of generated by the relations for any , then is isomorphic to the -algebra of continuous functions on the symmetric group , the latter interpreted as the subgroup of given by all permutation matrices. Hence, is a compact quantum supergroup of .
The free symmetric quantum groups are an easy family of compact matrix quantum groups, i.e., the intertwiner spaces of their corepresentation categories are induced by a category of partitions. More precisely, it is the category of all non-crossing partitions that induces the corepresentation categories of . The canonical generating set of partitions of is .