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