Generalized symmetric group

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 (C_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: C_m \cong \mu_m.

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 C_m \times C_2 (for m odd this is isomorphic to C_{2m}): the C_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 C_m (concretely, by taking the product of all the C_m values), while the sign map on the symmetric group yields the C_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 : :H_2(S(2k+1,n)) = \begin{cases} 1 & n \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