Generalized Symmetric Group

In mathematics, the generalized symmetric group is the wreath product $S(m,n) := Z_m \wr S_n$ of the cyclic group of order ''m'' and the symmetric group of order ''n''.

Examples

* For $m=1,$ the generalized symmetric group is exactly the ordinary symmetric group: $S(1,n) = S_n.$
* For $m=2,$ one can consider the cyclic group of order 2 as positives and negatives ($Z_2 \cong \$) and identify the generalized symmetric group $S(2,n)$ with the signed symmetric group.

Representation theory

There is a natural representation of elements of $S(m,n)$ as generalized permutation matrices, where the nonzero entries are ''m''-th roots of unity: $Z_m \cong \mu_m.$

The representation theory has been studied since ; see references in . As with the symmetric group, the representations can be constructed in terms of Specht modules; see .

Homology

The first group homology group (concretely, the abelianization) is $Z_m \times Z_2$ (for ''m'' odd this is isomorphic to $Z_$): the $Z_m$ factors (which are all conjugate, hence must map identically in an abelian group, since conjugation is trivial in an abelian group) can be mapped to $Z_m$ (concretely, by taking the product of all the $Z_m$ values), while the sign map on the symmetric group yields the $Z_2.$ These are independent, and generate the group, hence are the abelianization.

The second homology group (in classical terms, the Schur multiplier) is given by :
:$H_2(S(2k+1,n)) = \begin 1 & n < 4\\ \mathbf/2 & n \geq 4.\end$
:$H_2(S(2k+2,n)) = \begin 1 & n = 0, 1\\ \mathbf/2 & n = 2\\ (\mathbf/2)^2 & n = 3\\ (\mathbf/2)^3 & n \geq 4. \end$
Note that it depends on ''n'' and the parity of ''m:'' $H_2(S(2k+1,n)) \approx H_2(S(1,n))$ and $H_2(S(2k+2,n)) \approx H_2(S(2,n)),$ which are the Schur multipliers of the symmetric group and signed symmetric group.

References

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Permutation groups