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

Statement
It is possible to have a finite group $$G$$ and an irreducible representation $$\varphi$$ of $$G$$ over a splitting field in characteristic zero (and hence also over $$\mathbb{C}$$) such that, if $$\chi$$ denotes the character of $$\varphi$$, we have elements $$g_1, g_2 \in G$$ satisfying:


 * $$\chi(g_1) = \chi(g_2)$$, and
 * $$\varphi(g_1)$$ and $$\varphi(g_2)$$ are not similar matrices, i.e., they are not conjugate in the general linear group to which the representation maps $$G$$. In particular, they may have different characteristic polynomials.

Related facts

 * Character values up to permutation of characters need not determine automorphism class

Opposite facts

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

We see from the table above that the elements $$a$$ and $$x$$ have the same character value, namely $$0$$, but their images have different characteristic polynomials (the characteristic polynomial of $$a$$ is $$t^2 + 1$$, that of $$x$$ is $$t^2 - 1$$) and are hence the images are not similar matrices.