# Combinatorial problems that were solved using the representation theory of finite groups?

Question: What are some examples of problems in combinatorics that were solved using the representation theory of finite groups ?

I am aware the representation theory of finite groups plays a role in solving problems in number theory, for example automorphic forms. But what are some (important) examples from combinatorics.

1. We have the following paper from Diaconis: Group Representations in Probability and Statistics. In particular are the results related to card shuffling.
• IIRC, various instances of cyclic sieving have been proven using representation theory first, and only later (if at all) combinatorially. One survey is arxiv.org/abs/1008.0790 (but there is newer work around, too). – darij grinberg Aug 24 '18 at 16:00

In Bruce Sagan's book, $The$ $Symmetric$ $Group$ (second edition), he proves a unimodality theorem using representation theory---if I recall correctly, for the poset called $L(m,n)$---but this wasn't the first proof of that theorem (Corollary 5.4.10).