Divided Difference Operator for the Highest root Hessenberg variety

TL;DR

This paper introduces a new divided difference operator for the highest root Hessenberg variety, extending classical operators and confirming a recent conjecture through combinatorial and algebraic methods.

Contribution

It constructs a divided difference operator for Hessenberg varieties using GKM theory, generalizing classical operators and proving a specific conjecture.

Findings

Generalizes classical divided difference operators

Proves a special case of Shareshian and Wachs' conjecture

Uses combinatorial and algebraic methods based on root systems

Abstract

We construct a divided difference operator using GKM theory. This generalizes the classical divided difference operator for the cohomology of the complete flag variety. This construction proves a special case of a recent conjecture of Shareshian and Wachs. Our methods are entirely combinatorial and algebraic, and rely heavily on the combinatorics of root systems and Bruhat order.