Today, I was listening to someone give an exhausting proof of the fundamental theorem of algebra when I recalled that there was a short proof using Lie theory:
A finite extension $K$ of $\mathbb{C}$ forms a finite-dimensional vector space over $\mathbb{C}$, so the group of units $K^\times$ would be of the form $\mathbb{C}^n\setminus\{0\}$, which is simply connected for $n>1$. Since the operation on $K^\times$ is essentially just multiplication of polynomials over $\mathbb{C}$, it must be a Lie group. In sum, if $n>1$, then $K^\times$ is a simply connected abelian Lie group, thus isomorphic (as a Lie group) to $\mathbb{C}^n$, which is absurd (since $\mathbb{C}^n$ is torsion-free). Thus, $n=1$.
What other examples are there of theorems which yield such short or elegant proofs by appealing to Lie theory?
To clarify the criteria: I'm looking for (nontrivial) theorems that are usually stated in terms outside of Lie theory (e.g. the fundamental theorem of algebra) that can be proven in a particularly short or elegant way using Lie groups or Lie algebras.
Best Answer
The coefficients of the polynomial $$(1+x)(1+x^2)(1+x^3)\cdots(1+x^n)$$ are unimodal. This innocuous-looking fact is surprisingly hard to prove, and perhaps the most elegant proof uses the representation theory of semisimple Lie algebras. See Stanley's survey for further details and related examples.