Generalized symmetric group
In mathematics, the generalized symmetric group is the wreath product [math]\displaystyle{ S(m,n) := Z_m \wr S_n }[/math] of the cyclic group of order m and the symmetric group of order n.
Examples
- For [math]\displaystyle{ m=1, }[/math] the generalized symmetric group is exactly the ordinary symmetric group: [math]\displaystyle{ S(1,n) = S_n. }[/math]
- For [math]\displaystyle{ m=2, }[/math] one can consider the cyclic group of order 2 as positives and negatives ([math]\displaystyle{ Z_2 \cong \{\pm 1\} }[/math]) and identify the generalized symmetric group [math]\displaystyle{ S(2,n) }[/math] with the signed symmetric group.
Representation theory
There is a natural representation of elements of [math]\displaystyle{ S(m,n) }[/math] as generalized permutation matrices, where the nonzero entries are m-th roots of unity: [math]\displaystyle{ Z_m \cong \mu_m. }[/math]
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 [math]\displaystyle{ Z_m \times Z_2 }[/math] (for m odd this is isomorphic to [math]\displaystyle{ Z_{2m} }[/math]): the [math]\displaystyle{ Z_m }[/math] 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 [math]\displaystyle{ Z_m }[/math] (concretely, by taking the product of all the [math]\displaystyle{ Z_m }[/math] values), while the sign map on the symmetric group yields the [math]\displaystyle{ Z_2. }[/math] 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):
- [math]\displaystyle{ H_2(S(2k+1,n)) = \begin{cases} 1 & n \lt 4\\ \mathbf{Z}/2 & n \geq 4.\end{cases} }[/math]
- [math]\displaystyle{ 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} }[/math]
Note that it depends on n and the parity of m: [math]\displaystyle{ H_2(S(2k+1,n)) \approx H_2(S(1,n)) }[/math] and [math]\displaystyle{ H_2(S(2k+2,n)) \approx H_2(S(2,n)), }[/math] 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", J. London Math. Soc., 2 8 (4): 615–620, doi:10.1112/jlms/s2-8.4.615
- Can, Himmet (1996), "Representations of the Generalized Symmetric Groups", Contributions to Algebra and Geometry 37 (2): 289–307, http://www.emis.de/journals/BAG/vol.37/no.2/b37h2can.ps.gz
- Osima, M. (1954), "On the representations of the generalized symmetric group", Math. J. Okayama Univ. 4: 39–54
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