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 on n letters.

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 $S(m,n)$ as generalized permutation matrices, where the nonzero entries are mth 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