Boolean Schubert Structure Coefficients

TL;DR

This paper provides an explicit combinatorial formula for the equivariant Schubert structure constants for boolean elements across all Lie types, simplifying calculations especially in type A where constants are 0 or 1.

Contribution

It introduces a new explicit formula for Schubert structure constants for boolean elements, advancing understanding in algebraic geometry and combinatorics.

Findings

All Schubert structure constants for boolean elements in type A are 0 or 1.

The formula applies to all Lie types for boolean elements.

Provides a unified approach to compute structure constants explicitly.

Abstract

The Schubert problem asks for combinatorial models to compute structure constants of the cohomology ring with respect to Schubert classes and has been an important open problem in algebraic geometry and combinatorics that guided fruitful research for decades. In this paper, we provide an explicit formula for the (equivariant) Schubert structure constants $c_{uv}^w$ across all Lie types when the elements $u,v,w$ are boolean. In particular, in type $A$, all Schubert structure constants on boolean elements are either $0$ or $1$.