Question

If I have a matrix that has only one non-trivial symmetry and thus the irreducible representations of the symmetry group are two one-dimensional representations (the one that assigns 1 to both operations and the one that assigns one to the identity and -1 to the other symmetry operation) and if my symmetry group reduces in this case to $c_1 R_1 \oplus c_2 R_2$, where $R_1$ and $R_2$ are the two representations. What can I deduce about the number of eigenvalues and their degeneracies.