Generalized symmetric group
In mathematics, the generalized symmetric group is the wreath product of the cyclic group of order m and the symmetric group of order n.
Examples
- For the generalized symmetric group is exactly the ordinary symmetric group:
- For 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 : the factors can be mapped to, 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 :
Note that it depends on n and the parity of m: and which are the Schur multipliers of the symmetric group and signed symmetric group.