Generalized symmetric group
Encyclopedia
In mathematics
, the generalized symmetric group is the wreath product
of the cyclic group
of order m and the symmetric group
on n letters.
as generalized permutation matrices, where the nonzero entries are mth 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 module
s; see .
(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 :
Note that it depends on n and the sign of m:
and 
which are the Schur multipliers of the symmetric group and signed symmetric group.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the generalized symmetric group is the wreath product
Wreath product
In mathematics, the wreath product of group theory is a specialized product of two groups, based on a semidirect product. Wreath products are an important tool in the classification of permutation groups and also provide a way of constructing interesting examples of groups.Given two groups A and H...
of the cyclic groupCyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
of order m and the symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
on n letters.
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 as generalized permutation matrices, where the nonzero entries are mth 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 module
Specht module
In mathematics, a Specht module is one of the representations of symmetric groups studied by .They are indexed by partitions, and in characteristic 0 the Specht modules of partitions of n form a complete set of irreducible representations of the symmetric group on n points.-Definition:Fix a...
s; 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
Schur multiplier
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2 of a group G.It was introduced by in his work on projective representations.-Examples and properties:...
) is given by :
Note that it depends on n and the sign of m:
and which are the Schur multipliers of the symmetric group and signed symmetric group.