Generalized symmetric group explained
of the
cyclic group of order
m and the
symmetric group of order
n.
Examples
the generalized symmetric group is exactly the ordinary symmetric group:
one can consider the cyclic group of order 2 as positives and negatives (
) and identify the generalized symmetric group
with the
signed symmetric group.
Representation theory
There is a natural representation of elements of
as
generalized permutation matrices, where the nonzero entries are
m-th
roots of unity:
The representation theory has been studied since ; see references in . As with the symmetric group, the representations can be constructed in terms of Specht modules; see .
Homology
The first group homology group (concretely, the abelianization) is
(for
m odd this is isomorphic to
): the
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
(concretely, by taking the product of all the
values), while the sign map on the symmetric group yields the
These are independent, and generate the group, hence are the abelianization.
The second homology group (in classical terms, the Schur multiplier) is given by :
H2(S(2k+1,n))=\begin{cases}1&n<4\\
Z/2&n\geq4.\end{cases}
H2(S(2k+2,n))=\begin{cases}1&n=0,1\\
Z/2&n=2\\
(Z/2)2&n=3\\
(Z/2)3&n\geq4.
\end{cases}
Note that it depends on
n and the parity of
m: H2(S(2k+1,n)) ≈ H2(S(1,n))
and
H2(S(2k+2,n)) ≈ H2(S(2,n)),
which are the Schur multipliers of the symmetric group and signed symmetric group
