Towards combinatorial characterization of the smoothness of Hessenberg Schubert varieties

Abstract: A \emph{Hessenberg Schubert variety} is an irreducible component of the intersection of a Schubert variety and a Hessenberg variety, defined as the closure of a Schubert cell inside the Hessenberg variety. We consider the smoothness of Hessenberg Schubert varieties of regular semisimple Hessenberg varieties of type $A$ in this paper. We consider the smoothness of the intersection of a Schubert variety and a Hessenberg variety to ensure the smoothness of the corresponding Hessenberg Schubert variety. Specifically, we analyze the structure of the GKM graphs of the intersection of a Schubert variety indexed by some special permutations and a Hessenberg variety. The regularity of the GKM graph is completely characterized in terms of pattern avoidance, which is a necessary (and also sufficient conjecturally) condition for the intersection to be smooth. We then extend the pattern avoidance result to all permutations, which is believed to be a sufficient condition for the corresponding Hessenberg Schubert variety to be smooth.