# Character value need not determine similarity class of image under irreducible representation

## Statement

It is possible to have a finite group and an irreducible representation of over a splitting field in characteristic zero (and hence also over ) such that, if denotes the character of , we have elements satisfying:

- , and
- and are not similar matrices, i.e., they are not conjugate in the general linear group to which the representation maps . In particular, they may have different characteristic polynomials.

## Related facts

### Opposite facts

- Character determines representation in characteristic zero: In other words, at a
*global*level, the character value does determine the similarity class.

## Proof

### Example of the dihedral group

`Further information: faithful irreducible representation of dihedral group:D8`

Consider dihedral group:D8 (see also linear representation theory of dihedral group:D8) and its faithful irreducible two-dimensional representation.

The dihedral group of order eight has a two-dimensional irreducible representation, where the element acts as a rotation (by an angle of ), and the element acts as a reflection about the first axis. The matrices are:

This particular choice of matrices give a representation as orthogonal matrices, and in fact, the representation is as signed permutation matrices (i.e., it takes values in the signed symmetric group of degree two). Thus, it is also a monomial representation.

Below is a description of the matrices based on the above choice as well as another formulation involving complex unitary matrices:

Element | Matrix (orthogonal/monomial/signed permutation matrices) | Matrix as complex unitary | Characteristic polynomial | Minimal polynomial | Trace, character value | Determinant |
---|---|---|---|---|---|---|

2 | 1 | |||||

0 | 1 | |||||

-2 | 1 | |||||

0 | 1 | |||||

0 | -1 | |||||

0 | -1 | |||||

0 | -1 | |||||

0 | -1 | |||||

Set of values used | -- | -- | ||||

Ring generated by values used (characteristic zero) | -- ring of integers | -- ring of Gaussian integers | -- | -- | -- ring of integers | -- ring of integers |

Field generated by values used (characteristic zero) | -- field of rational numbers | -- | -- | -- field of rational numbers | -- field of rational numbers |

We see from the table above that the elements and have the same character value, namely , but their images have different characteristic polynomials (the characteristic polynomial of is , that of is ) and are hence the images are not similar matrices.