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

2016/04/15 by Lee, Seung Jin
#05E15 #14N15 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1604.04523

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.

Related

• Conjectural positivity of Chern-Schwartz-MacPherson classes for Richardson cells
• Structure constants for Chern classes of Schubert cells
• Positivity of Chern classes of Schubert cells and varieties
• Pieri and Murnaghan--Nakayama type Rules for Chern classes of Schubert Cells
• Chevalley formulae for the motivic Chern classes of Schubert cells and for the stable envelopes