Paper · Mar 4, 2022

Product structure and regularity theorem for totally nonnegative flag varieties

Huanchen Bao, Xuhua He

math.RTarXiv:2203.02137
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Abstract

The totally nonnegative flag variety was introduced by Lusztig. It has enriched combinatorial, geometric, and Lie-theoretic structures. In this paper, we introduce a (new) -total positivity on the full flag variety of an arbitrary Kac-Moody group, generalizing the (ordinary) total positivity. We show that the -totally nonnegative flag variety has a cellular decomposition into totally positive -Richardson varieties. Moreover, each totally positive -Richardson variety admits a favorable decomposition, called a product structure. Combined with the generalized Poincare conjecture, we prove that the closure of each totally positive -Richardson variety is a regular CW complex homeomorphic to a closed ball. Moreover, the -total positivity on the full flag provides a model for the (ordinary) totally nonnegative partial flag variety. As a consequence, we prove that the closure of each (ordinary) totally positive Richardson variety is a regular CW complex homeomorphic to a closed ball, confirming conjectures of Galashin, Karp and Lam.

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Submitted
Mar 4, 2022
Subjects
math.RT, math.AG, math.CO, math.GN

Comments: 30 pages

Poincaré conjecture

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  1. Ricci Flow and the Poincare Conjecture

    John W. Morgan and Gang TianThe canon

    Topics Poincaré conjecture

  2. On 4-dimensional smooth Poincare conjecture

    Selman Akbulut

    Topics Poincaré conjecture

  3. Equations in Products of Free Groups and 3-Manifold Groups, I

    Olga Kharlampovich and Alina Vdovina

    Topics Poincaré conjecture · Shared terms “homeomorphic”, “combinatorial”

  4. Moving Manifolds and the Poincare Conjecture

    David V. Svintradze

    Topics Poincaré conjecture

  5. A Lean Formalization of the Hamilton–Perelman Proof of the Three-Dimensional Poincaré Conjecture

    Ziyang Qin, Yuan Liao, Ayush Khaitan et al.

    Topics Poincaré conjecture

  6. Rigor with Machine Learning from Field Theory to the Poincaré Conjecture

    Sergei Gukov, James Halverson and Fabian Ruehle

    Topics Poincaré conjecture

  7. Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture

    Huai-Dong Cao and Xi-Ping ZhuThe canon

    Topics Poincaré conjecture

  8. Differential topology of the spaces of asymptotically stable vector fields and Lyapunov functions

    Matthew D. Kvalheim

    Topics Poincaré conjecture · Shared terms “generalized Poincare”

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Poincaré conjecture Every closed, simply connected 3-manifold is homeomorphic to the 3-sphere. Poincaré posed the question in 1904, and Perelman proved it in 2002–03… Start with: Generalized Poincare's Conjecture in Dimensions Greater Than Four (Smale, 1961)

In the guidebook

  1. 7A.12
    The Poincaré Conjecture, PreciselyBook 7A · Topology and the Fundamental Group

    Every word of the statement, and why each hypothesis is needed.

  2. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.

  3. 10A.6
    Necks, Caps and Their TopologyBook 10A · Three-Manifolds and Geometrization

    What an ε-neck is, how necks overlap and chain into tubes, and which manifolds necks and caps can build.