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Modified symmetric group

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

Definition

As a matrix group

For every $N\in \N$ the modified symmetric group for dimension $N$ is the subgroup of the 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 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 $N$ .

Table of Contents

Modified symmetric group

Definition

As a matrix group

As a direct product

Basic properties

Representation theory

Cohomology

Related quantum groups

References

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Table of Contents

Modified symmetric group

Definition

As a matrix group

As a direct product

Basic properties

Representation theory

Cohomology

Related quantum groups

References

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