Generalized symmetric group

Template:Short description 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 Template:Harv; see references in Template:Harv. As with the symmetric group, the representations can be constructed in terms of Specht modules; see Template:Harv.

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 Template:Harv:

$H_2(S(2k+1,n))=\{\begin{array}{cc}1& n<4\\ 𝐙/2& n\ge 4.\end{array}$

$H_2(S(2k+2,n))=\{\begin{array}{cc}1& n=0,1\\ 𝐙/2& n=2\\ (𝐙/2{)}^2& n=3\\ (𝐙/2{)}^3& n\ge 4.\end{array}$

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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