Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Intersection multiplicity one for the Belkale-Kumar product in G/B

Nicolas Ressayre, Luca Francone

TL;DR

This work proves that the Belkale-Kumar product ${\odot_0}$ on $H^*(G/B)$ has structure constants $c_{uv}^w$ equal to either $0$ or $1$, specifically showing $c_{uv}^w=1$ whenever $\Phi(u)\cap\Phi(v)=\Phi(w)$ and $\Phi(u)\cup\Phi(v)=\Phi^+$. The proof combines a geometric reformulation via incidence varieties with a root-system combinatorial theorem, reducing the problem to linear-algebra statements about tangent maps and kernels controlled by a matrix $M$. Consequences span the BRUhat order, descent statistics, LR cone regular faces, and cohomological tensor-product components, providing a uniform, case-free treatment across all types. The results connect geometric incidence, birationality, and eigencone geometry with deep root-system combinatorics and yield a uniform framework for understanding cohomology product structures on flag varieties.

Abstract

Consider the complete flag variety $X$ of a complex semisimple algebraic group $G$. We show that the structure coefficients of the Belkale-Kumar product $\odot_0$, on the cohomology $\mathrm{H}^{*}(X,\mathbf{Z})$, are all either $0$ or $1$. We also derive some consequences. The proof that is mainly geometric also uses new combinatorial results on root systems. Moreover, it is uniform and avoids case by case considerations.

Intersection multiplicity one for the Belkale-Kumar product in G/B

TL;DR

This work proves that the Belkale-Kumar product on has structure constants equal to either or , specifically showing whenever and . The proof combines a geometric reformulation via incidence varieties with a root-system combinatorial theorem, reducing the problem to linear-algebra statements about tangent maps and kernels controlled by a matrix . Consequences span the BRUhat order, descent statistics, LR cone regular faces, and cohomological tensor-product components, providing a uniform, case-free treatment across all types. The results connect geometric incidence, birationality, and eigencone geometry with deep root-system combinatorics and yield a uniform framework for understanding cohomology product structures on flag varieties.

Abstract

Consider the complete flag variety of a complex semisimple algebraic group . We show that the structure coefficients of the Belkale-Kumar product , on the cohomology , are all either or . We also derive some consequences. The proof that is mainly geometric also uses new combinatorial results on root systems. Moreover, it is uniform and avoids case by case considerations.
Paper Structure (27 sections, 30 theorems, 79 equations)

This paper contains 27 sections, 30 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Some consequences of Theorem \ref{['th:main']}
  3. On the Bruhat order
  4. The number of descents
  5. Using a Belkale-Kumar expression of cuvw
  6. Minimal regular faces of the eigencone
  7. Cohomological components of tensor products
  8. Overview of the proof of Theorem \ref{['th:main']}
  9. The geometric strategy
  10. Incidence variety
  11. Incidence variety as a fibered product
  12. The tangent map of eta
  13. About birational maps
  14. First properties of the map eta
  15. Bruhat order
  16. ...and 12 more sections

Key Result

Theorem 1

Let $u,v$ and $w$ in $W$ be such that $\Phi(u)\cap\Phi(v)=\Phi(w)$ and $\Phi(u)\cup\Phi(v)=\Phi^+$. Then

Theorems & Definitions (51)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Corollary 5
  • Corollary 6
  • Theorem 7: Dimitrov-Roth
  • Corollary 8
  • Proposition 9
  • Proposition 10
  • ...and 41 more