====== Free symmetric quantum group ======
By a **free symmetric quantum group** one means any element of the one-parameter sequence $(S_N^+)_{N\in \N}$ of [[compact matrix quantum group|compact matrix quantum groups]] defined by Wang in [(:ref:Wang98)] under the name //quantum permutation groups//. Each $S_N^+$ is a [[free_orthogonal_easy_quantum_groups|free]] counterpart of the [[wp>symmetric group]] $S_N$ of the corresponding dimension $N$. 

===== Definition =====

Given $N\in \N$, the **free symmetric quantum group** $S_N^+$ (or **quantum permuation group on** $N$ **symbols**) is the [[compact matrix quantum group]] $(C(S_N^+),u)$ where $u=(u_{i,j})_{i,j=1}^N$ organizes the generators $\{u_{i,j}\}_{i,j=1}^N$ of the (unital) [[wp>Universal_C*-algebra|universal C*-algebra]] 
$$C(S_N^+)\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,\forall_{i,j=1}^N:u_{i,j}^2=u_{i,j}=u_{i,j}^\ast, \, {\textstyle\sum_{k=1}^N} u_{i,k}={\textstyle\sum_{l=1}^N} u_{l,j}=1\big\rangle,$$
where $1$ is the unit of the universal $C^\ast$-algebra.

In other words, the entries $\{u_{i,j}\}_{i,j=1}^N$ of the fundamental corpresentation matrix $u$ of $S_N^+$ 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 $1$ (where the orthogonality is to mean  $u_{i,j}u_{i,k}=\delta_{j,k}u_{i,j}$ and $u_{i,j}u_{l,j}=\delta_{i,l}u_{i,j}$ for all $i,j,k,l=1,\ldots,N$ as can be shown). Those relations are commonly summarized by saying that $u$ is a **magic unitary**.

===== Basic Properties =====

The fundamental corepresentation matrix $u$ of $S_N^+$ is in particular //orthogonal//. Hence, $S_N^+$ is a compact quantum subgroup of the [[free orthogonal quantum group]] $O_N^+$.

If $I$ denotes the closed two-sided ideal of $C(S_N^+)$ generated by the relations $u_{i,j}u_{k,l}=u_{k,l}u_{i,j}$ for any $i,j,k,l=1,\ldots, N$, then $C(S_N^+)/I$ is isomorphic to the $C^\ast$-algebra $C(S_N)$ of continuous functions on the [[wp>symmetric group]] $S_N$, the latter interpreted as the subgroup of $\mathrm{GL}(N,\C)$ given by all [[wp>permutation matrix|permutation matrices]]. Hence, $S_N^+$ is a compact quantum supergroup of $S_N$.

The free symmetric quantum groups $(S_N^+)_{N\in \N}$ are an [[easy_quantum_group|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 $\mathrm{NC}$ of [[category_of_all_non-crossing_partitions|all non-crossing partitions]] that induces the corepresentation categories of $(S_N^+)_{N\in \N}$. The canonical generating set of partitions of $\mathrm{NC}$ is $\{\fourpart,\singleton\}$. 
===== Representation theory =====
===== Cohomology =====
===== Related quantum groups =====

===== References =====

[( :ref:Wang98 >>
author   : Shuzhou Wang
title    : Quantum Symmetry Groups of Finite Spaces
journal  : Communications in Mathematical Physics
year     : 1998
volume   : 195
number   : 1
pages    : 195--211
url      : http://dx.doi.org/10.1007/s002200050385
archivePrefix: arXiv
eprint   :0707.3168
)]