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On two notions of total positivity for generalized partial flag varieties of classical Lie types

Grant Barkley, Jonathan Boretsky, Christopher Eur, Jiyang Gao

Abstract

For Grassmannians, Lusztig's notion of total positivity coincides with positivity of the Plucker coordinates. This coincidence underpins the rich interaction between matroid theory, tropical geometry, and the theory of total positivity. Bloch and Karp furthermore characterized the (type A) partial flag varieties for which the two notions of positivity similarly coincide. We characterize the symplectic (type C) and odd-orthogonal (type B) partial flag varieties for which Lusztig's total positivity coincides with Plucker positivity.

On two notions of total positivity for generalized partial flag varieties of classical Lie types

Abstract

For Grassmannians, Lusztig's notion of total positivity coincides with positivity of the Plucker coordinates. This coincidence underpins the rich interaction between matroid theory, tropical geometry, and the theory of total positivity. Bloch and Karp furthermore characterized the (type A) partial flag varieties for which the two notions of positivity similarly coincide. We characterize the symplectic (type C) and odd-orthogonal (type B) partial flag varieties for which Lusztig's total positivity coincides with Plucker positivity.

Paper Structure

This paper contains 29 sections, 37 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Pinnings
  3. Generalities and notations
  4. Pinnings of Sp2n and SO2n+1
  5. Type C pinning
  6. Type B pinning
  7. Compatibility of pinnings
  8. Partial flag varieties
  9. Bruhat orders
  10. Lusztig positivity and the proof of (3)=>(1)=>(2)
  11. Lusztig's total positivity
  12. Proof of (1)=>(2)
  13. Proof of (3)=>(1)
  14. Examples, non-examples, and the proof of (2)=>(3)
  15. Proof of (2)=>(3)
  16. ...and 14 more sections

Key Result

Theorem 1.1

blochkarp2023 The following are equivalent for a subset $K\subseteq[n-1]$:

Theorems & Definitions (94)

  • Theorem 1.1
  • Theorem A
  • Definition 2.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • proof
  • Remark 3.5
  • Definition 3.6
  • ...and 84 more