First a technical definition.

Let . Let , for and . For , let where . This defines an action of on and hence on the set of equivalence classes of , . We call two representations , *-equivalent*, and write

if and for some . Similarly, we call two characters , of * -equivalent*, and write

if for some .

For example, Let , and . Then .

Let be as in the previous post. Note that

for . Therefore, the matrix representations of two -equivalent representations differ only

by a character.

Let .

The results in the above section tells us how to construct all the irreducible representations of . We must

- write down all the characters (i.e., 1-dimensional representations) of ,
- describe the action of on ,
- for each , compute the stabilizer ,
- describe all irreducible representations of each ,
- write down the formula for the character of .

Write as , where each component is a character of the cyclic group , . Let denote all the distinct characters which occur in , so

Let denote the number of ‘s in , denote the number of ‘s in , …, denote the number of ‘s in . Then . Call this the *partition associated to *.

If two characters , belong to the same class in , under the -equivalence relation, then their associated partitions are equal.

The Frobenius formula for the character of an induced representation gives the following character formula. Let denote the character of . Then

for all and . In particular, if then