While thinking about something else, I noticed recently the following result, which is certainly not new:
Let $latex G$ be a compact topological group [ADDITIONAL ASSUMPTION pointed out by Y. Choi: connected, Lie group], and let $latex \rho$ be a finite-dimensional irreducible unitary continuous representation of $latex G$ on a vector space $latex V$. Then the natural representation $latex \pi$ of $latex G$ on $latex \mathrm{End}(V)$ decomposes as a direct sum of one-dimensional characters if and only if $latex \rho$ is of dimension $latex 1$.
One direction is clear: if $latex \rho$ has dimension one, then $latex \pi$ is simply the trivial one-dimensional representation. For the converse, here is an argument with character theory.
As a first step, note that if $latex \rho$ (of dimension $latex d\geq 1$, say) has this property, then in fact $latex \pi$ decomposes as a direct sum of distinct one-dimensional characters: indeed, the multiplicity of a character $latex \chi$ in $latex \pi$ is the same as
$latex n_{\chi}=\int_{G}\chi(x)\mathrm{Tr}(\pi(g))dg,$
where $latex dg$ is the probability Haar measure on $latex G$, and since
$latex \mathrm{Tr}(\pi(g))=|\mathrm{Tr}(\rho(g))|^2,$
we get
$latex n_{\chi}\leq \int_{G}\mathrm{Tr}(\pi(g))dg=1$
by the orthogonality relations of characters. (Algebraically, this is just an application of Schur’s lemma).
Thus if we decompose $latex \pi$ into irreducible representations, we get
$latex \pi=\bigoplus_{1\leq i\leq d^2} \chi_i,$
where the $latex \chi_i$ are distinct one-dimensional characters. We then know by orthogonality that
$latex d^2=\int_{G} |\mathrm{Tr}(\pi(g))|^2 dg=\int_{G} |\mathrm{Tr}(\rho(g))|^4 dg.$
Now the last-integral is bounded by
$latex \int_{G} |\mathrm{Tr}(\rho(g))|^4 dg\leq \mathrm{Max}_{g}|\mathrm{Tr}(\rho(g))|^2 \times \int_G|\mathrm{Tr}(\rho(g))|^2dg\leq d^2,$
(since $latex |\mathrm{Tr}(\rho(g))|\leq d$). Comparing, this means that there must be equality throughout in this estimate, which in turn implies that $latex |\mathrm{Tr}(\rho(g))|=d$ for all $latex g\in G$. Since $latex \rho(g)$ is unitary of size $latex d$, this implies that $latex \rho(g)$ is scalar for all $latex g$, and since it is assumed to be irreducible, it is in fact one-dimensional.
I see two interesting points in this argument: (1) is there a purely algebraic proof of the last part? I haven’t thought very hard about this yet, but it would be nice to have one; (2) the appearance of the fourth moment of $latex \rho$ is nicely reminiscent of the Larsen alternative (see Section 6.3 of my notes on representation theory, for instance…)
