Last update: 20 August 2013
Here, we use the fact that in the symmetric group is conjugate to
Let be a partition of The conjugacy class of labeled by consists of all permutations with cycle structure i.e., permutations having one-cycles, two-cycles, etc. We count by first writing down an element of this class. There are ways of doing this, since there are ways of listing the numbers. Now consider the in this permutation. There are cyclic permutations of length of this cycle all of which give the same permutation. Moreover, there are ways of reordering the different so
Thus orthogonality of characters for the symmetric group can be written as
We now return to the representation theory of From the definition (2.13) of the weighted trace wtr of we have The trace tr is the trace of some representation of so it is a character of and can be written in terms of the irreducible as The coefficient is a non-negative integer known as a Kostka coefficient. Thus, we have We know that Denote it by Then we have This holds for all with We make the assumption that so that the matrix is square. From (2.32) and and (1.58), we have Therefore, we have or In other words, where is the number of partitions of
Definition 2.38 The hyper octahedral group is the group of by matrices such that
|(1)||there is exactly one nonzero entry in each row and each column,|
|(2)||this nonzero entry is either or|
Homework Problem 2.39
|(2)||can be given by generators and relations|
|(3)||Compute the characters of (Hint: you need to use two alphabets.)|
Thus is a signed permutation, and so we have Therefore, we have proved Frobenius formula up to sign. That is, we have shown To prove that the sign must be positive, we again consider the representation theory of We know that where is the irreducible indexed by In particular so if we can show that then the Frobenius formula is proved. From (2.31), we have so we are interested in a formula for multiplying power symmetric functions and Schur functions. With this in mind, we make some definitions.
Definition 2.40 If are partitions, the we say that if for all
Example 2.41 If and then We picture this as The whole picture is the partition and the boxes that are not dotted make up the partition
Definition 2.42 If then the skew diagram is the set-theoretic difference of the Ferrers diagrams of and
In example 2.41, the dotted boxes form Two boxes in are connected if they share a common edge.
Definition 2.43 A border strip is a skew diagram that
|(1)||contains no two-by-two blocks of boxes, and|
|(2)||is connected; that is, consists of a single sequence of connected boxes.|
Example 2.44 The dotted boxes in the following diagram form a border strip of length 8. The skew diagram in Example 2.41 does not form a border strip, because it is not connected. The dotted boxes in do not form a border strip, because they contain a two-by-two block of boxes.
This is a copy of lectures in Representation Theory given by Arun Ram, compiled by Tom Halverson, Rob Leduc and Mark McKinzie.