====== Modified symmetric group ======

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

===== Definition =====

==== As a matrix group ====

For every $N\in \N$ the **modified 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 multiplied by a factor of $1$ or $-1$, i.e., the set
$$S_N\colon\hspace{-0.6em}=\{ u\in \C^{N\times N}\,\vert\, \exists r\in \{-1,1\},\,\forall_{i,j=1}^N: u_{i,j}\in\{0,1\},\,{\textstyle\sum_{\ell=1}^N} u_{i,\ell}={\textstyle\sum_{k=1}^N} u_{k,j}=r\},$$
where $u=(u_{i,j})_{i,j=1}^N$.

==== As a direct product ====


The modified symmetric group for dimension $N$, where $N\in \N$, can also be defined as the [[group direct product|direct product]] of groups $S_N'\colon\hspace{-0.66em}=\Z_2\times S_N$ of the cyclic group $\Z_2\equiv \Z/2\Z$ of order $2$ and the [[symmetric group]] for dimension $N$.


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

The modified 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]]. More precisely, it is the [[category of partitions of even size]] that induces the corepresentation categories of $(S_N')_{N\in \N}$. Its canonical generating set of partitions is $\{\crosspart,\fourpart,\singleton\otimes\singleton\}$. 

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

===== Cohomology =====

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

===== References =====