Equivariant Combinatorics

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The term "Equivariant Combinatorics" was coined at a summer 2017 conference in Montreal. An equivariant map is a map that intertwines with symmetries. In combinatorics, which is the study of discrete objects, it can also mean that the map intertwines certain statistics on combinatorial objects or the representations associated to them. Broadly, it describes profound interactions between algebraic combinatorics, algebraic geometry, algebraic topology, representation theory, and theoretical physics because these subjects are wedded to the study of symmetries in nature and mathematics. The phrase therefore also encompasses ties to probability and semigroup theory through representation theory. The main premise of this project is the exploration and further development of these new relationships. This project also has a substantial computational component (contributing to and using the rapidly growing open source mathematical software SageMath).The project is aimed at solving five open equivariant combinatorics problems. They will be attacked with various teams of collaborators and include: (1) The study of a crystal basis explaining the plethysm in the character theory of the general linear group. This has applications to physics, invariant theory and complexity theory. (2) A combinatorial analysis of LLT polynomials, which are important symmetric functions in combinatorics and geometry. This involves new statistics on the underlying combinatorial objects. (3) A proof of the Shimozono-Weyman conjecture. This relies on a sign-reversing involution compatible with katabolism. (4) Crystals associated to Lie superalgebras. This requires developing the theory of irreducible crystals for Lie superalgebras that originated in string theory. (5) A unified theory of Tsetlin libraries. This is a new approach to the theory of Markov chains associated to semigroups.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.