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Poset polytopes and pipe dreams: toric degenerations and beyond

Ievgen Makedonskyi, Igor Makhlin

Abstract

We demonstrate how pipe dreams can be applied to the theory of poset polytopes to produce toric degenerations of flag varieties. Specifically, we present such constructions for marked chain-order polytopes of Dynkin types A and C. These toric degenerations also give rise to further algebraic and geometric objects such as PBW-monomial bases and Newton--Okounkov bodies. We discuss a construction of the former in the type A case and of the latter in type C.

Poset polytopes and pipe dreams: toric degenerations and beyond

Abstract

We demonstrate how pipe dreams can be applied to the theory of poset polytopes to produce toric degenerations of flag varieties. Specifically, we present such constructions for marked chain-order polytopes of Dynkin types A and C. These toric degenerations also give rise to further algebraic and geometric objects such as PBW-monomial bases and Newton--Okounkov bodies. We discuss a construction of the former in the type A case and of the latter in type C.
Paper Structure (12 sections, 13 theorems, 16 equations)

This paper contains 12 sections, 13 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Type A
  3. Poset polytopes
  4. Pipe dreams
  5. Toric degenerations
  6. PBW-monomial bases
  7. Type C
  8. Type C poset polytopes
  9. Type C pipe dreams
  10. The intermediate Schubert degeneration
  11. Toric degenerations
  12. Newton--Okounkov bodies

Key Result

Theorem 0

Fix $O\subset P$. For every order ideal $J$ one can choose $M_J\subset P$ and $k_J\in\mathbb N$ so that the map $\psi:X_J\mapsto X_{w_{M_J}(1),\dots,w_{M_J}(k_J)}$ is an isomorphism and $\psi(I_O)$ is an initial ideal of $I$. Consequently, the toric variety of $\mathcal{Q}_O(\lambda)$ is a flat dege

Theorems & Definitions (21)

  • Theorem 0: cf. Theorem \ref{['degenmain']}
  • Definition 1
  • Proposition 1
  • Proposition 2
  • Definition 2
  • Proposition 3
  • Theorem 1
  • proof : Sketch of proof
  • Theorem 2
  • Definition 3
  • ...and 11 more