Towards combinatorial characterization of the smoothness of Hessenberg Schubert varieties

TL;DR

The work tackles the problem of determining when Hessenberg Schubert varieties $\Omega_{w,h}$ are smooth inside regular semisimple Hessenberg varieties in type $A$ by analyzing the GKM graphs of intersections with $\mathrm{Hess}(S,h)$. It develops the $h$-Bruhat order and an edge-injection on the associated GKM graphs $\Gamma_{w,h}$, proving that vertex degrees nondecrease along the order and identifying seven associated patterns whose avoidance in $w\in\mathcal{G}_h$ guarantees regularity (and hence smoothness of the intersection); for arbitrary $w$, it extends the criterion by using the corresponding $\widetilde{w}\in\mathcal{G}_h$ and four additional patterns, yielding a ten-pattern condition. The main contributions are (i) a concrete combinatorial criterion linking pattern avoidance to the regularity of $\Gamma_{w,h}$, (ii) a method to transfer regularity from a representative $\widetilde{w}$ to general $w$, and (iii) conjectures that these pattern-avoidance conditions are also equivalent to smoothness of the Hessenberg Schubert varieties $\Omega_{w,h}$. This establishes a bridge between combinatorial pattern theory and geometric smoothness in Hessenberg settings, with potential implications for cohomology representations and torus-fixed point structures.

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.

Figures (5)

• Figure 1: The incomparability graph of $h=(3,4,4,4)$
• Figure 2: The induced subgraph $\Gamma_{w,h}$ of $\Gamma_{h}$ for a Hessenberg function $h=(3,3,4,4)$ and $w=2134\in\mathcal{G}_h$.
• Figure 3: Seven associated patterns and induced subgraphs on $\{i,j,k,\ell\}$.
• Figure 4: Four additional associated patterns and the induced subgraph on $\{i,j,k,\ell,m\}$.
• Figure 5: Relations between the regularity of $\Gamma_{w,h}$ and the smoothness of $\Omega_{w,h}$

Theorems & Definitions (63)

• Theorem 1.1
• Theorem A: Theorem \ref{['thm:interval']}
• Theorem B: Theorem \ref{['thm:increasing']}
• Theorem C: Theorems \ref{['thm:irregular']}, \ref{['thm:regular']} and \ref{['thm:main']}
• Proposition 2.1
• Lemma 2.2
• Proposition 2.3: Chain Property
• Remark 2.4
• Proposition 2.5: T2
• Proposition 2.6: CHL, CHL2, HP