Combinatorial Representation Theory of Lie Algebras. Richard Stanley’s Work and the Way It Was Continued

Representation theory is a fundamental tool for studying group symmetry − geometric, analytic, or algebraic − by means of linear algebra, which has important applications to other areas of mathematics and mathematical physics. One very successful trend in this field in recent decades involves using combinatorial objects to model the representations, which allows one to apply combinatorial methods for studying them, e.g., for concrete computations. This trend led to the emergence of combinatorial representation theory, which has now become a thriving area. Richard Stanley played a crucial role, through his work and his students, in the development of this new area. In the early stages, he has the merit to have pointed out to combinatorialists, in [34, 36], the potential that representation theory has for applications of combinatorial methods. Throughout his distinguished career, he wrote significant articles which touch upon various combinatorial aspects related to representation theory (of Lie algebras, the symmetric group, etc.). I will describe some of Richard’s contributions involving Lie algebras, as well as recent developments inspired by them (including some open problems), which attest the lasting impact of his work.