= Solution
The <character of a representation> is $\chi_V(g)=\operatorname{tr}\rho_V(g)$. Average any positive inner product over the compact group using <Haar measure> to obtain a <unitary representation>. On the <maximal torus> $\operatorname{diag}(z,z^{-1})$, the commuting unitary operators admit a simultaneous eigenbasis. Each eigenline carries a circle <character of a representation> $z^n$ by part (a). Hence
$$
\boxed{\chi_V(z)=\sum_{n\in\mathbb Z}m_nz^n,\qquad m_n\in\mathbb N_0,}
$$
with finitely many nonzero coefficients. Every element of <special unitary group> is conjugate to an element of this torus, so this describes the full <character of a representation> via its torus restriction. Here the coefficient notation includes zero multiplicities.
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