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](http://www-math.mit.edu/~rstan/pubs/) (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.



Modern sources on Lie algebraic methods in combinatorics
* A lot of the literature on crystals (in the modern sense of algebraic combinatorics) uses the theory of crystal bases (which is part of quantum group theory) as a black box. For one example of such a use, see [Mark Shimozono, *Crystals for dummies*](http://www.aimath.org/WWN/kostka/crysdumb.pdf), which proves various properties of semistandard Young tableaux using crystals.
* Vertex algebras can be used to prove partition identities. For example: [Debajyoti Nandi, *Partition identities arising from the standard $A\_2^{(2)}$-modules of level $4$*, thesis](https://rucore.libraries.rutgers.edu/rutgers-lib/45379/) or [Mirko Primc, *Vertex Algebras and Combinatorial Identities*](http://link.springer.com.ololo.sci-hub.ac/article/10.1023/A:1019747408807) and [S. Capparelli, *On Some Representations of Twisted Affine Lie Algebras and Combinatorial Identities*](https://doi.org/10.1006/jabr.1993.1017). Do you count vertex algebras as part of Lie-algebra theory?
* [Jean-Louis Loday, Todor Popov, *Parastatistics Algebra, Young Tableaux and the Super Plactic Monoid*](http://www-irma.u-strasbg.fr/~loday/PAPERS/2008LodayPopov%28Parastat%29.pdf) obtains a generalized Cauchy identity for hook Schur functions (a super-version of Schur functions) from representations of Lie superalgebras.


