Alon and Kozma re-proved Kirchoff’s matrix-tree theorem using representation theory of $S_n$:
Let $G$ be any weighted graph, and denote by $w_e$ the weight of the edge $e$. For a spanning tree $T$, denote $w(T) = \prod_{e \in T} w_e$, where the product is over all edges $e$ of $T$. Finally denote by $0 = \lambda_0 \le \lambda_1 \le \cdots \le \lambda_{n-1}$ the eigenvalues of the continuous time Laplacian $\Delta_G$. Then $$\sum_{T} w(T) = \frac{1}{n} \prod_{i=1}^{n-1} \lambda_i.$$
It was actually a corollary of a result on some mixing properties of the interchange process (a natural occurring random walk on $S_n$).
See the Appendix of their paper.
Of course, the old theorem has many classical proofs, but representation theory gives a new angle.