The question is given below:
And the hint at the back of the book says:
Establish the corresponding equality for characters.
And this was a question I was helped on it, which establish the relation between characters and $\Phi_{n}$: Question 4, chapter III, section 7 in Vinberg "Linear representations of groups. "
So, I end up with having $$ tr {\Phi_{m}} tr{\Phi_{n}} = \frac{z^{m+1} - z^{-m-1}}{z-z^{-1}} .\frac{z^{n+1} - z^{-n-1}}{z-z^{-1}}$$ but then what, could anyone help me in establishing the above relation mentioned in the question?
Expand out one of the terms your expression:
$$\frac{z^{m+1} - z^{-m-1}}{z-z^{-1}} .\frac{z^{n+1} - z^{-n-1}}{z-z^{-1}} = \frac{z^{m+1} - z^{-m-1}}{z-z^{-1}} (z^n+z^{n-2}+\dots +z^{-n})$$
Do the multiplication:
$$= \frac{z^{m+n+1}+z^{m+n-1}+\dots+z^{m-n+1} - z^{-m+n-1} - z^{-m+n-3} \dots - z^{-m-n-1}}{z-z^{-1}}$$
Rearrange and collect pairs of terms from outside to inside:
$$= \frac{(z^{m+n+1}- z^{-m-n-1}) + (z^{m+n-1} - z^{-m-n+1})+\dots+(z^{m-n+1} - z^{-m+n-1})}{z-z^{-1}}$$