On tangent cones to Schubert varieties in type $E$

TL;DR

This paper investigates tangent cones to Schubert varieties in type E groups, showing that certain pairs of involutions lead to distinct tangent cones, thus revealing geometric differences in these algebraic structures.

Contribution

It proves that for good pairs of involutions in the Weyl group of types E6, E7, and E8, the tangent cones to their Schubert varieties are distinct as subschemes.

Findings

Tangent cones to Schubert varieties are distinct for good involution pairs.

The result applies specifically to types E6, E7, and E8.

Distinct tangent cones imply geometric differences in Schubert varieties.

Abstract

We consider tangent cones to Schubert subvarieties of the flag variety $G/B$, where $B$ is a Borel subgroup of a reductive complex algebraic group $G$ of type $E_6$, $E_7$ or $E_8$. We prove that if $w_1$ and $w_2$ form a good pair of involutions in the Weyl group $W$ of $G$ then the tangent cones $C_{w_1}$ and $C_{w_2}$ to the corresponding Schubert subvarieties of $G/B$ do not coincide as subschemes of the tangent space to $G/B$ at the neutral point.