Characteristic cycles of IH sheaves of simply laced minuscule Schubert varieties are irreducible

TL;DR

This paper proves that the characteristic cycles of intersection homology sheaves of Schubert varieties in certain complex flag manifolds are irreducible, using computational methods, completing a classification for exceptional Lie types.

Contribution

It establishes the irreducibility of IH sheaves for Schubert varieties in type E6 and E7, extending previous results to exceptional Lie types with computational verification.

Findings

Characteristic cycles are irreducible for the studied cases.

Mather classes and Chern-Schwartz-MacPherson classes are strongly positive.

The proof combines an algorithm for local Euler obstructions with computer calculations.

Abstract

Let $G/P$ be a complex cominuscule flag manifold of type $E_6,E_7$. We prove that each characteristic cycle of the intersection homology (IH) complex of a Schubert variety in $G/P$ is irreducible. The proof utilizes an earlier algorithm by the same authors which calculates local Euler obstructions, then proceeds by direct computer calculation using Sage. This completes to the exceptional Lie types the characterization of irreducibility of IH sheaves of Schubert varieties in cominuscule $G/P$ obtained by Boe and Fu. As a by-product, we also obtain that the Mather classes, and the Chern-Schwartz-MacPherson classes of Schubert cells in cominuscule $G/P$ of type $E_6,E_7$, are strongly positive.