Towards combinatorial characterization of the smoothness of Hessenberg Schubert varieties

TL;DR

This paper characterizes the smoothness of Hessenberg Schubert varieties in type A by analyzing GKM graph regularity and pattern avoidance, providing a combinatorial criterion for smoothness.

Contribution

It introduces a pattern avoidance criterion that fully characterizes the smoothness of Hessenberg Schubert varieties in type A.

Findings

Smoothness corresponds to GKM graph regularity

Pattern avoidance is necessary and sufficient for smoothness

Provides a combinatorial criterion for smoothness

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 intersected with 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 and a Hessenberg variety. Our results show that the regularity of these GKM graphs is completely characterized in terms of pattern avoidance, which is a necessary and sufficient condition for the intersection to be smooth. This shows that our pattern avoidance provides a sufficient condition for the smoothness of a Hessenberg Schubert variety.