Let $G$ be a compact connected Lie group, let $T\le G$ be a maximal torus, and fix a choice of positive roots $R^+$ for the root system of the complexified [Lie algebra](/page/Lie%20Algebra) of $G$ with respect to $T$. Let $\Lambda^+$ denote the corresponding set of dominant integral weights in the character lattice $X^*(T)$. Let $V$ and $V'$ be finite-dimensional complex vector spaces, and let

\begin{align*} \rho:G\to GL(V) \end{align*}

and

\begin{align*} \rho':G\to GL(V') \end{align*}

be continuous irreducible complex representations of $G$. Define their characters by

\begin{align*} \chi_V(g):=\operatorname{tr}(\rho(g)) \end{align*}

and

\begin{align*} \chi_{V'}(g):=\operatorname{tr}(\rho'(g)) \end{align*}

for $g\in G$. If the highest weights of $V$ and $V'$ with respect to the fixed pair $(T,R^+)$ are equal to the same weight $\lambda\in\Lambda^+$, then

\begin{align*} \chi_V=\chi_{V'} \end{align*}

as functions $G\to\mathbb C$. Consequently, $V$ and $V'$ are isomorphic as complex representations of $G$.