Workshop on Combinatorics of Enumerative Geometry

Abstract: There are two major research trends in the theory of symmetric functions arising from Dyck paths. One is the theory of Catalan symmetric functions and its geometric realization conceived by Chen-Haiman, following the works of Broer and Shimozono-Weyman. The symmetric function part of this story was established by a series of works of Blasiak-Morse-Pun-Summers. Another is the study of chromatic symmetric functions of graphs, that utilize the fact that Dyck paths roughly correspond to unit interval graphs. We first exhibit a family of smooth algebraic varieties that realize Catalan symmetric functions, and explain how it imply the geometric predictions of Broer, Shimozono-Weyman, Chen-Haiman, and Blasiak-Morse-Pun. We then exhibit that our varieties also represents the chromatic symmetric functions of unit interval graphs. Thus, our varieties might be understood as a geometric realization of the world of Dyck paths (which are related but not the same as the Hessenberg varieties). If time and situation allows, we will present some more recent works on this subject. This talk is based on arXiv:2301.00862 and arXiv:2410.12231.