Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Order polynomial product formulas and poset dynamics

Sam Hopkins

Abstract

We survey all known examples of finite posets whose order polynomials have product formulas, and we propose the heuristic that these are the same posets with good dynamical behavior. Here the dynamics in question are the actions of promotion on the linear extensions of the poset and rowmotion on the P-partitions of the poset.

Order polynomial product formulas and poset dynamics

Abstract

We survey all known examples of finite posets whose order polynomials have product formulas, and we propose the heuristic that these are the same posets with good dynamical behavior. Here the dynamics in question are the actions of promotion on the linear extensions of the poset and rowmotion on the P-partitions of the poset.

Paper Structure

This paper contains 42 sections, 20 theorems, 25 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A more detailed account of the heuristic
  3. Whence all this?
  4. The open problems
  5. Acknowledgments
  6. The posets
  7. Shapes and shifted shapes
  8. Root posets
  9. Minuscule posets
  10. The "chain of V's"
  11. Order polynomial product formulas
  12. Symmetry classes of plane partitions
  13. Minuscule posets
  14. Root posets
  15. Doppelgängers
  16. ...and 27 more sections

Key Result

Theorem 3.1

Let $P$ be a minuscule poset. Then where $r\colon P\to \mathbb{N}$ is the rank function of $P$.

Figures (3)

  • Figure 1: Examples of the families of shapes.
  • Figure 2: Left: some crystallographic root posets. Right: the non-crystallographic root posets of coincidental type.
  • Figure 3: Left: the other minuscule posets besides the rectangle and shifted staircase. Right: the "chain of V's."

Theorems & Definitions (28)

  • Theorem 3.1: Proctor proctor1984bruhat
  • Theorem 3.2: Cellini--Papi cellini2002adnilpotent, Haiman haiman1994conjectures
  • Theorem 3.3: proctor1983trapezoidproctor1990newwilliams2013cataland
  • Theorem 3.4: proctor1983trapezoid
  • Theorem 3.5: Proctor
  • Theorem 3.6: Kreweras--Niederhausen kreweras1981solution
  • Theorem 3.7: Hopkins--Lai hopkins2020plane; Okada okada2020intermediate
  • Proposition 4.1: Schützenberger schutzenberger1972promotion
  • Theorem 4.2
  • Theorem 4.3: Haiman haiman1989mixedhaiman1992dual
  • ...and 18 more