generalized symmetric group

• 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 ($Z\_2\; \backslash cong\; \backslash \{\backslash pm\; 1\backslash \}$) and identify the generalized symmetric group $S(2,n)$ with the signed symmetric group.

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: $Z\_m\; \backslash cong\; \backslash mu\_m.$

The representation theory has been studied since {{Harv|Osima|1954}}; see references in {{Harv|Can|1996}}. As with the symmetric group, the representations can be constructed in terms of Specht modules; see {{Harv|Can|1996}}.

The first group homology group – concretely, the abelianization – is $Z\_m\; \backslash times\; Z\_2$ (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 {{Harv|Davies|Morris|1974}}:

:

\mathbf{Z}/2 & n \geq 4.\end{cases}

:

\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: $H\_2(S(2k+1,n))\; \backslash approx\; H\_2(S(1,n))$ and $H\_2(S(2k+2,n))\; \backslash approx\; H\_2(S(2,n)),$ which are the Schur multipliers of the symmetric group and signed symmetric group.

{{reflist}}

{{refbegin}}

• {{citation | first1 = J. W. | last1 = Davies | first2 = A. O. | last2 = Morris | title = The Schur Multiplier of the Generalized Symmetric Group | year = 1974 | journal = J. London Math. Soc. | series = 2 | volume = 8 | issue = 4 | pages = 615–620 |doi=10.1112/jlms/s2-8.4.615 }}
• {{citation | first = Himmet | last = Can | title = Representations of the Generalized Symmetric Groups | journal = Contributions to Algebra and Geometry | volume = 37 | year = 1996 | number = 2 | pages = 289–307 | url = http://www.emis.de/journals/BAG/vol.37/no.2/b37h2can.ps.gz |citeseerx=10.1.1.11.9053}}
• {{citation | first = M. | last = Osima | title=On the representations of the generalized symmetric group | journal = Math. J. Okayama Univ. | volume = 4 | year = 1954 | pages = 39–54}}

{{refend}}

Category:Permutation groups