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The Ehrhart $h^*$-polynomials of positroid polytopes

Yuhan Jiang

Abstract

A positroid is a matroid realized by a matrix such that all maximal minors are non-negative. Positroid polytopes are matroid polytopes of positroids. In particular, they are lattice polytopes. The Ehrhart polynomial of a lattice polytope counts the number of integer points in the dilation of that polytope. The Ehrhart series is the generating function of the Ehrhart polynomial, a rational function with a numerator called the $h^*$-polynomial. We give explicit formulas for the $h^*$-polynomials of an arbitrary positroid polytope regarding permutation descents. Our result generalizes that of Early, Kim, and Li for hypersimplices.

The Ehrhart $h^*$-polynomials of positroid polytopes

Abstract

A positroid is a matroid realized by a matrix such that all maximal minors are non-negative. Positroid polytopes are matroid polytopes of positroids. In particular, they are lattice polytopes. The Ehrhart polynomial of a lattice polytope counts the number of integer points in the dilation of that polytope. The Ehrhart series is the generating function of the Ehrhart polynomial, a rational function with a numerator called the -polynomial. We give explicit formulas for the -polynomials of an arbitrary positroid polytope regarding permutation descents. Our result generalizes that of Early, Kim, and Li for hypersimplices.
Paper Structure (15 sections, 25 theorems, 34 equations, 11 figures)

This paper contains 15 sections, 25 theorems, 34 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Organization
  3. Acknowledgements
  4. Positroids
  5. Grassmann necklaces
  6. Positroid polytopes
  7. Connected positroids
  8. Circuit triangulation of connected positroid polytopes
  9. The $h^*$-polynomial of connected positroid polytopes
  10. The $h^*$-polynomial of half-open connected positroid polytopes
  11. Tree positroids
  12. Positroid tiling of positroid polytopes
  13. decorated permutations, plabic graphs
  14. Decorated permutations
  15. Plabic graphs

Key Result

Theorem 1.1

Let $P_\mathcal{J}$ be any connected positroid polytope (see def:connected), where $\mathcal{J}$ is the associated Grassmann necklace (see def:Grassneck). Let $D_\mathcal{J} \subset S_n$ be the subset of permutations that label the circuit triangulation of $P_\mathcal{J}$ (see thm:trian). For any $w where $\mathrm{cover}(w)$ is the number of elements $w$ covers in the poset $\mathcal{P}_{w_0,\math

Figures (11)

  • Figure 1: We show the graph of the circuit triangulation of the positroid polytope $P_\mathcal{J}$ associated to the positroid with Grassmann necklace $\mathcal{J} = (123,235,345,145,125)$, which coincides with the Hasse diagram of the poset $\mathcal{P}_{24135,\mathcal{J}}$. The $h^*$-polynomial of $P_\mathcal{J}$ is $1+4z+3z^2$.
  • Figure 2: The positroid polytope $P_\mathcal{J}$ associated to the Grassmann necklace $(12,23,13,14)$, with bases $\{12,13,14,23,24\}$.
  • Figure 3: On the left, we show the graph $\Gamma_{2,5}$ of the circuit triangulation of the hypersimplex $\Delta_{2,5}$. The vertices of $\Gamma_{2,5}$ are labeled by permutations $w \in D_{3,5}$ in one-line notation. On the right, we relabel the vertices $w$ of $\Gamma_{2,5}$ by affine permutations $L_{w_0}(w)$ in window notation with $w_0 = 31425$ according to \ref{['lem:relabel']}. The arrows represent cover relations in the poset $\mathcal{P}_{31425,\mathcal{J}}$ for $\mathcal{J} = (12,23,34,45,51)$, pointing from a smaller element to a bigger element.
  • Figure 4: We show the graph of the circuit triangulation of the positroid polytope associated with the Grassmann necklace $\mathcal{J} = (124, 234, 134, 145, 125)$. The vertices of $\Gamma_\mathcal{J}$ are labeled by $D_\mathcal{J}$.
  • Figure 5: The positroid polytope associated to the Grassmann necklace $\mathcal{J} = (12,23,13,14)$ is a pyramid. The red facet corresponds to $F_1: x_1 = 1$; the yellow facet corresponds to $F_2: x_2 = 1$; the blue facet corresponds to $F_3: x_1+x_2+x_3 = 2$. These are all the upper facets of this positroid polytope.
  • ...and 6 more figures

Theorems & Definitions (82)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4: postnikov2006, Lem 16.3
  • Theorem 2.5: postnikov2006oh2017
  • Example 2.6
  • Definition 2.7
  • Proposition 2.8: positroidspolyposi
  • Definition 2.9
  • ...and 72 more