Character value need not determine similarity class of image under irreducible representation
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.
- Character determines representation in characteristic zero: In other words, at a global level, the character value does determine the similarity class.
Example of the dihedral group
Further information: faithful irreducible representation of dihedral group:D8
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|
|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.