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.
  • $\begingroup$ 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). $\endgroup$ – 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).

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