Conjectural positivity of Chern-Schwartz-MacPherson classes for Richardson cells

TL;DR

The paper conjectures that Chern-Schwartz-MacPherson classes of Richardson cells in flag varieties have nonnegative coefficients, linking this to a new product with sign-alternating structure constants, and discusses implications for positivity.

Contribution

It introduces a conjecture on the positivity of CSM classes for Richardson cells and relates it to a new product with sign-alternating structure constants, supported by computational evidence.

Findings

Conjecture on nonnegative coefficients of CSM classes for Richardson cells.

Connection between the positivity conjecture and the sign behavior of a new product’s structure constants.

Computational evidence motivating the conjecture.

Abstract

Following some work of Aluffi-Mihalcea-Sch\"{u}rmann-Su for the CSM classes of Schubert cells and some elaborate computer calculations by R. Rimanyi and L. Mihalcea, I conjecture that the CSM classes of the Richardson cells expressed in the Schubert basis have nonnegative coefficients. This conjecture was principally motivated by a new product $\square$ coming from the Segre classes in the cohomology of flag varieties (such that the associated Gr of this product is the standard cup product) and the conjecture that the structure constants of this new product $\square$ in the standard Schubert basis have alternating sign behavior. I prove that this conjecture on the sign of the structure constants of $\square$ would follow from my above positivity conjecture about the CSM classes of Richardson cells.