====== Symmetric group ======

A **symmetric group** is any member of a certain sequence $(S_N)_{N\in \N}$ of [[classical orthogonal matrix groups]].

===== Definition =====

For every $N\in \N$ the **symmetric group** for dimension $N$ is the subgroup of the [[wp>general linear group]] $\mathrm{GL}(N,\C)$ given by all permutation $N\times N$-matrices, i.e., the set
$$S_N\colon\hspace{-0.6em}=\{ u\in \C^{N\times N}\,\vert\, \forall_{i,j=1}^N: u_{i,j}\in\{0,1\},\,{\textstyle\sum_{\ell=1}^N} u_{i,\ell}={\textstyle\sum_{k=1}^N} u_{k,j}=1\},$$
where $u=(u_{i,j})_{i,j=1}^N$.



===== Basic properties =====

The symmetric 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]]. In fact, the [[category of all partitions]] induces the corepresentation categories of $(S_N)_{N\in \N}$. It is canonically generated by  $\{\crosspart,\fourpart, \singleton\}$. 

===== Representation theory =====

===== Cohomology =====

===== Related quantum groups =====

===== References =====