Modern sources on Lie algebraic methods in combinatorics

Question

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.

Answer 1

• 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, 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 or Mirko Primc, Vertex Algebras and Combinatorial Identities and S. Capparelli, On Some Representations of Twisted Affine Lie Algebras and Combinatorial Identities. 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 obtains a generalized Cauchy identity for hook Schur functions (a super-version of Schur functions) from representations of Lie superalgebras.