This proposal aims to investigate several aspects of the interplay between combinatorics and representation theory. First, the PI will investigate the enumerative cyclic sieving phenomenon introduced by Reiner, Stanton, and White as it applies to combinatorial actions on the cluster complexes of Fomin and Zelevinsky as well as actions on tableaux arising from the theory of crystals. The PI will use objects from representation theory such as Kazhdan-Lusztig bases and cluster algebras to investigate these combinatorial problems. Second, the PI will investigate connections between hyperplane arrangements and the Hilbert series of certain rings related to the diagonal coinvariant module.
The enumeration of objects in a finite set is the fundamental problem of combinatorics and has many scientific applications outside of pure mathematics. Symmetry groups were introduced by the ancient Greeks and are the primary motivation for the study of representation theory. So, the study of enumeration up to symmetry is of paramount importance in combinatorial representation theory and algebraic combinatorics. This project uses representation theory to indirectly count the elements of finite sets, yielding a deeper and more elegant understanding of their enumeration.
