I am looking for modern textbooks, expository papers and journal articles regarding the use of Lie algebras in combinatorics. In particular, I am interested to know the extent to which people still care about this.

For example, the representation theory of $\mathfrak{sl}_2(\mathbf{C})$ can be used to show that certain sequences are unimodal and symmetric. This and similar material can be found in Richard Stanley's papers (e.g. 41, 57, 62, 72, 84).

In the textbooks I've looked at (e.g. Aigner's *A Course in Enumeration*, Stanley's EC1/2) you can find material on things like posets, Young tableaux, and the character theory of $S_n$ and $\mathrm{GL}(n)$. But it seems like Lie algebras such as $\mathfrak{sl}_2(\mathbf{C})$ have been left out.

Crystals for dummiesdoes a lot of tableau combinatorics based on using the theory of crystal bases as a black box, which he gives references for. As far as I understand, at least the majority of these references uses Lie-algebraic methods to prove things. $\endgroup$ – darij grinberg Jul 9 '17 at 12:59Vertex Algebras and Combinatorial Identitiesand S. Capparelli,On Some Representations of Twisted Affine Lie Algebras and Combinatorial Identities. $\endgroup$ – darij grinberg Jul 9 '17 at 13:06