TITLE: Applications of Representation Theory in Combinatorics
QUESTION [21 upvotes]: What are the examples of interesting combinatorial identities (e.g. bijection between two sets of combinatorial objects) that can be proved using representation theory, or has some representation theoretic interpretation. Please also mention whether a purely combinatorial proof is known in each case.

REPLY [3 votes]: Here's an example that involves a group other than the symmetric group.
Let ${\mathbb F}_{q^n}$ denote the finite field with $q^n$ elements.  Since ${\mathbb F}_{q^n}$ is a vector space of dimension $n$ over ${\mathbb F}_q$, we may regard a generator of the multiplicative group ${\mathbb F}^*_{q^n}$ as an element of $GL_n({\mathbb F}_q)$; define a Singer cycle to be such a generator.  Also define a reflection to be a nontrivial element of $GL_n({\mathbb F}_q)$ whose fixed points form a hyperplane.

Theorem (Lewis–Reiner–Stanton). The number of factorizations of a Singer cycle into a product of $n$ reflections is $(q^n-1)^{n-1}$.

The only known proof uses the character theory of $GL_n({\mathbb F}_q)$ in an essential way.
Note that the above theorem is analogous to the classical result that the number of factorizations of an $n$-cycle in ${\mathfrak S}_n$ into a product of $n-1$ transpositions is $n^{n-2}$.