Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
Last update: 31 March 2012
The Hyperoctahedral Group
The hyperoctahedral group is the group of signed permutations of i.e. bijections
There are multiple notations for signed permutations
Two line notations: where in the second line is below in the first line, for
In cycle notation as permutations in the symmetric group
Matrix notation: where the
entry is if is positive and if is negative, and all other entries are
Diagram notation: where the dot in the top row is connected to the th dot in the bottom row and the edge is labeled by if is negative.
The hyperoctahedral group is also called the Weyl group of type and the Weyl group of type and is the same as the group
It is the group of matrices such that
there is exactly one nonzero entry in each row and each column,
each nonzero entry is
The group is isomorphic to the wreath product
and has order
The reflections in are the elements
The simple reflections are
The hyperoctahedral group can be presented by generators
There are three things to show:
The simple reflections in satisfy the given relations.
The simple reflections in generate
The group given by generators and the relations in the statement has
The following pictures show that
satisfy the relations in the statement of the theorem.
For each let
Every can be written as
where is the permutation given by
Since generate and
for it follows that generate
Let be the free group generated by modulo the relations in the statement of the theorem. We will show that every element is either
Let and assume that
First we will show that every element of can be written in the form
Then, by the induction assumption,
It follows that if
Then, by the induction assumption
Let be a pair of partitions sucht that the total number of boxes in and is A standard tableau of shape is a filling of the boxes of and with such that, in each partition,
the rows of are increasing left to right,
the columns of are increasing top to bottom.
The rows and columns of each partition are indexed as for matrices,
The numbers and are the content and the sign of the box respectively.
The irreducible representations of the hyperoctahedral group are indexed by pairs of partitions with boxes total.
of standard tableaux of shape
The irreducible module is given by
with basis and with action given by
is the content of the box containing in
is the sign of the box containing in
is the same as except that and are switched, and
if is not a standard tableau.
We must show that
The are modules,
The are irreducible modules,
The are inequivalent modules,
These are all the simple modules.
The proofs are similar to the proofs of the analogous statements in the symmetric group case.
Define elements in the group algebra of the hyperoctahedral group by
The action of these elements of the module is given by
where and are the sign and the content of the box containing in respectively.
where is the number of standard tableaux of shape Thus is the dimension of the irreducible module for the symmetric group Then