This is an old revision of the document!
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 and where is the transpose of .