From motivic Chern classes of Schubert cells to their Hirzebruch and CSM classes

TL;DR

This paper studies motivic Chern classes of Schubert cells in flag manifolds, exploring their specializations, dualities, and relations to Hirzebruch and CSM classes, with conjectures on positivity and recursion formulas.

Contribution

It establishes new properties and dualities of motivic Chern classes, connects them to Hirzebruch and CSM classes, and formulates conjectures on their positivity and combinatorial structure.

Findings

Specializations of motivic Chern classes at y=-1 and y=0

Leading terms of motivic classes yield CSM classes

Recursion formulas for Hirzebruch classes of Schubert cells

Abstract

The equivariant motivic Chern class of a Schubert cell in a `complete' flag manifold $X=G/B$ is an element in the equivariant K theory ring of $X$ to which one adjoins a formal parameter $y$. In this paper we prove several `folklore results' about the motivic Chern classes, including finding specializations at $y=-1$ and $y=0$; the coefficient of the top power of $y$; how to obtain Chern-Schwartz-MacPherson (CSM) classes as leading terms of motivic classes; divisibility properties of the Schubert expansion of motivic Chern classes. We collect several conjectures about the positivity, unimodality, and log concavity of CSM and motivic Chern classes of Schubert cells, including a conjectural positivity of structure constants of the multiplication of Poincar\'e duals of CSM classes. In addition, we prove a `star duality' for the motivic Chern classes. We utilize the motivic Chern transformation to define two equivariant variants of the Hirzebruch transformation, which appear naturally in the Grothendieck-Hirzebruch-Riemann-Roch formalism. We utilize the Demazure-Lusztig recursions from the motivic Chern class theory to find similar recursions giving the Hirzebruch classes of Schubert cells, their Poincar{\'e} duals, and their Segre versions. We explain the functoriality properties needed to extend the results to `partial' flag manifolds $G/P$.