Another exercise with characters
While thinking about something else, I noticed recently the following result, which is certainly not new:
Let be a compact topological group [ADDITIONAL ASSUMPTION pointed out by Y. Choi: connected, Lie group], and let be a finite-dimensional irreducible unitary continuous representation of on a vector space . Then the natural representation of on decomposes as a direct sum of one-dimensional characters if and only if is of dimension .
One direction is clear: if has dimension one, then is simply the trivial one-dimensional representation. For the converse, here is an argument with character theory.
As a first step, note that if (of dimension , say) has this property, then in fact decomposes as a direct sum of distinct one-dimensional characters: indeed, the multiplicity of a character in is the same as
where is the probability Haar measure on , and since
by the orthogonality relations of characters. (Algebraically, this is just an application of Schur’s lemma).
Thus if we decompose into irreducible representations, we get
where the are distinct one-dimensional characters. We then know by orthogonality that
Now the last-integral is bounded by
(since ). Comparing, this means that there must be equality throughout in this estimate, which in turn implies that for all . Since is unitary of size , this implies that is scalar for all , 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 is nicely reminiscent of the Larsen alternative (see Section 6.3 of my notes on representation theory, for instance…)