Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Paolo Aluffi, Leonardo C. Mihalcea, Jörg Schürmann, Changjian Su

TL;DR

The paper develops a unified framework for motivic Chern classes of Schubert cells in flag varieties, connecting them to Hirzebruch and Chern-Schwartz-MacPherson classes through Demazure-Lusztig recursions and a star duality. It formulates two equivariant Hirzebruch transformations (unnormalized and normalized) and shows how DL operators induce recursive computations of Hirzebruch and Segre versions, aligning with Hecke algebra actions. A key achievement is identifying CSM classes as leading terms of motivic Chern classes under a special substitution for the parameter $y$, and extending these constructions to partial flags $G/P$ with robust functoriality. The work also collects and motivates positivity, unimodality, and log-concavity conjectures for CSM and motivic Chern classes, and outlines an interpretation in the Hecke algebra context, with star duality furnishing a powerful symmetry between motivic and dual classes. Together, these results provide computational tools and theoretical bridges between K-theory, cohomology, and representation-theoretic structures in Schubert calculus."

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 Poincaré 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{é} duals, and their Segre versions. We explain the functoriality properties needed to extend the results to `partial' flag manifolds $G/P$.

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

TL;DR

The paper develops a unified framework for motivic Chern classes of Schubert cells in flag varieties, connecting them to Hirzebruch and Chern-Schwartz-MacPherson classes through Demazure-Lusztig recursions and a star duality. It formulates two equivariant Hirzebruch transformations (unnormalized and normalized) and shows how DL operators induce recursive computations of Hirzebruch and Segre versions, aligning with Hecke algebra actions. A key achievement is identifying CSM classes as leading terms of motivic Chern classes under a special substitution for the parameter , and extending these constructions to partial flags with robust functoriality. The work also collects and motivates positivity, unimodality, and log-concavity conjectures for CSM and motivic Chern classes, and outlines an interpretation in the Hecke algebra context, with star duality furnishing a powerful symmetry between motivic and dual classes. Together, these results provide computational tools and theoretical bridges between K-theory, cohomology, and representation-theoretic structures in Schubert calculus."

Abstract

The equivariant motivic Chern class of a Schubert cell in a `complete' flag manifold is an element in the equivariant K theory ring of to which one adjoins a formal parameter . In this paper we prove several `folklore results' about the motivic Chern classes, including finding specializations at and ; the coefficient of the top power of ; 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 Poincaré 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{é} duals, and their Segre versions. We explain the functoriality properties needed to extend the results to `partial' flag manifolds .
Paper Structure (31 sections, 36 theorems, 268 equations)

This paper contains 31 sections, 36 theorems, 268 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Equivariant (co)homology
  4. Equivariant K theory
  5. The Chern character
  6. Operators in equivariant cohomology and $\mathrm{K}$ theory of flag manifolds
  7. Schubert data
  8. BGG, Demazure, and Demazure-Lusztig operators
  9. Leading terms of DL operators
  10. Equivariant motivic Chern classes
  11. Preliminaries about motivic Chern classes
  12. Motivic Chern classes of Schubert cells
  13. Integrals of motivic Chern classes and point counting
  14. Divisibility properties
  15. The parabolic case
  16. ...and 16 more sections

Key Result

Theorem 1.1

(cf. prop:HDL-rec) Let $W$ be the Weyl group of $G$, $w \in W$, and $s_i$ a simple reflection such that $ws_i >w$ in the Bruhat ordering. Consider the Schubert cell $X(w)^\circ \subseteq G/B$, of dimension $\ell(w)$. Then the Hirzebruch classes are determined by the following recursions:

Theorems & Definitions (87)

  • Theorem 1.1
  • Remark 3.1
  • Proposition 3.2: lusztig:eqK
  • Lemma 3.3
  • Lemma 3.4
  • Proposition 3.5
  • proof
  • Remark 4.1
  • Theorem 4.2
  • Remark 4.3
  • ...and 77 more