The question and its answer is given below:
Where T is the unit circle, and $\Phi_{n}$ is described below:
But I do not understand the solution,could anyone explain it for me please? or give me understandable solution as I am stucked in this problem.
A basis of $\Phi_n$ is given by the monomials $f_m=u_1^mu_2^{n-m}$, and for $A(z)=\mathrm{diag}(z,z^{-1})$, we have $ \Phi_n(A(z))(f_m)=z^m(z^{-1})^{n-m}f_m$. Therefore $$\mathrm{tr}\,\Phi_n(A(z))=\chi_n(A(z))=z^n+z^{n-1}z^{-1}+\cdots zz^{-n+1}+z^{-n},$$ which is equal to $(z^{n+1}-z^{-n-1})/(z-z^{-1})$.