Young diagrams and intersection numbers for toric manifolds associated with Weyl chambers

TL;DR

This paper provides a combinatorial formula for intersection numbers in toric manifolds linked to Weyl chambers, revealing the cohomology ring structure for classical and exceptional root systems.

Contribution

It introduces a new combinatorial approach to compute intersection numbers in toric manifolds associated with Weyl chambers, including exceptional types.

Findings

Derived explicit formulas for intersection numbers

Connected intersection numbers to Weyl group elements

Enhanced understanding of cohomology ring structures

Abstract

We study intersection numbers of invariant divisors in the toric manifold associated with the fan determined by the collection of Weyl chambers for each root system of classical type and of exceptional type $G_2$. We give a combinatorial formula for intersection numbers of certain subvarieties which are naturally indexed by elements of the Weyl group. These numbers describe the ring structure of the cohomology of the toric manifold.