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 ( $Z_{2}\cong \{\pm 1\}$ ) 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 (Osima 1954); see references in (Can 1996). As with the symmetric group, the representations can be constructed in terms of Specht modules; see (Can 1996).

Homology

The first group homology group (concretely, the abelianization) is $Z_{m}\times Z_{2}$ (for m odd this is isomorphic to $Z_{{2m}}$ ): 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 (Davies & Morris 1974):

H_{2}(S(2k+1,n))={\begin{cases}1&n<4\\{\mathbf {Z}}/2&n\geq 4.\end{cases}}

H_{2}(S(2k+2,n))={\begin{cases}1&n=0,1\\{\mathbf {Z}}/2&n=2\\({\mathbf {Z}}/2)^{2}&n=3\\({\mathbf {Z}}/2)^{3}&n\geq 4.\end{cases}}

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

Davies, J. W.; Morris, A. O. (1974), "The Schur Multiplier of the Generalized Symmetric Group" (PDF), J. London Math. Soc., 2, 8: 615–620
Can, Himmet (1996), "Representations of the Generalized Symmetric Groups", Contributions to Algebra and Geometry, 37 (2): 289–307
Osima, M. (1954), "On the representations of the generalized symmetric group", Math. J. Okayama Univ., 4: 39–54

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