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Total positivity in twisted product of flag varieties

Huanchen Bao, Xuhua He

Abstract

We show that the totally nonnegative part of the twisted product of flag varieties of a Kac-Moody group admits a cellular decomposition, and the closure of each cell is a topological manifold with boundary. We also establish explicit parameterizations of each totally positive cell. In the special cases of double flag varieties and braid varieties, we show that the totally nonnegative parts are regular CW complexes homeomorphic to closed balls. Moreover, we prove that the link of any totally nonnegative double Bruhat cell in a reductive group is a regular CW complex homeomorphic to a closed ball, solving an open problem of Fomin and Zelevinsky.

Total positivity in twisted product of flag varieties

Abstract

We show that the totally nonnegative part of the twisted product of flag varieties of a Kac-Moody group admits a cellular decomposition, and the closure of each cell is a topological manifold with boundary. We also establish explicit parameterizations of each totally positive cell. In the special cases of double flag varieties and braid varieties, we show that the totally nonnegative parts are regular CW complexes homeomorphic to closed balls. Moreover, we prove that the link of any totally nonnegative double Bruhat cell in a reductive group is a regular CW complex homeomorphic to a closed ball, solving an open problem of Fomin and Zelevinsky.
Paper Structure (31 sections, 22 theorems, 79 equations, 1 figure)

This paper contains 31 sections, 22 theorems, 79 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Flag varieties
  3. Main result
  4. Special case: braid varieties
  5. Special case: double flag varieties
  6. Special case: double Bruhat cells
  7. Our strategy
  8. Flag varieties and their totally nonnegative parts
  9. Kac-Moody groups
  10. Flag varieties
  11. Expressions and subexpressions
  12. Totally nonnegative flag varieties
  13. Twisted product of flag varieties
  14. Monoid structure on $W$
  15. The thickening map
  16. ...and 16 more sections

Key Result

Theorem 1

[Theorem thm:Zp, Proposition prop:TM$\&$ Proposition prop:duality] (1) Each stratum $\mathcal{Z}\xspace_{v, \ul{w}, >0}$ is a topological cell and we have an explicit parameterization of the cell; (2) The closure of each totally positive stratum in $\mathcal{Z}\xspace$ is a union of some totally pos

Figures (1)

  • Figure 1: $\tilde{G}$ for $SL_4$

Theorems & Definitions (38)

  • Theorem 1
  • Theorem 2: Theorem \ref{['thm:braid']}
  • Example 1.1
  • Theorem 3: Theorem \ref{['thm:double']}
  • Theorem 4: Theorem \ref{['thm:link']}
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Definition 3.1
  • Proposition 3.2
  • ...and 28 more