Chern class of Schubert cells in the flag manifold and related algebras

TL;DR

This paper explores the relationship between Chern-Schwartz-MacPherson classes of Schubert cells, algebraic structures like Fomin-Kirillov and nil-Hecke algebras, and their positivity properties in flag manifolds.

Contribution

It establishes a connection between nonnegativity conjectures in algebra and positivity of characteristic classes in geometry, providing new insights and proofs.

Findings

Nonnegativity conjecture in Fomin-Kirillov algebra implies positivity of Chern-Schwartz-MacPherson classes.

Chern-Schwartz-MacPherson classes are expressed as sums of structure constants of equivariant cohomology.

Refined positivity conjectures are proposed based on algebraic and geometric insights.

Abstract

We discuss a relationship between Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds, Fomin-Kirillov algebra, and the generalized nil-Hecke algebra. We show that nonnegativity conjecture in Fomin-Kirillov algebra implies the nonnegativity of the Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds for type A. Motivated by this connection, we also prove that the (equivariant) Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds are certain summations of the structure constants of the equivariant cohomology of the Bott-Samelson varieties. We also discuss the refined positivity conjectures of the Chern-Schwartz-MacPherson classes for Schubert cells motivated by the nonnegativity conjecture in Fomin-Kirillov algebra.