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The omega invariant of a matroid

Alex Fink, Kris Shaw, David E Speyer

Abstract

The third author introduced the $g$-polynomial $g_M(t)$ of a matroid, a covaluative matroid statistic which is unchanged under series and parallel extension. The $g$-polynomial of a rank $r$ matroid $M$ has the form $g_1 t + g_2 t^2 + \cdots + g_r t^r$. The coefficient $g_1$ is Crapo's classical $β$-invariant. In this paper, we study the coefficient $g_r$, which we term the $ω$-invariant of $M$. We show that, if $M/F$ is connected for every proper flat $F$ of $M$, and $ω(N)$ is nonnegative for every minor $N$ of $M$, then all the coefficients of $g_M(t)$ are nonnegative. We give several simplified versions of Ferroni's formula for $ω(M)$, and compute $ω(M)$ when $r$ or $|E(M)|-2r$ is small.

The omega invariant of a matroid

Abstract

The third author introduced the -polynomial of a matroid, a covaluative matroid statistic which is unchanged under series and parallel extension. The -polynomial of a rank matroid has the form . The coefficient is Crapo's classical -invariant. In this paper, we study the coefficient , which we term the -invariant of . We show that, if is connected for every proper flat of , and is nonnegative for every minor of , then all the coefficients of are nonnegative. We give several simplified versions of Ferroni's formula for , and compute when or is small.

Paper Structure

This paper contains 22 sections, 62 theorems, 83 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Piecewise polynomials and Minkowski weights
  5. The ring of piecewise polynomial functions
  6. Minkowski weights
  7. Tautological classes of a matroid and the Bergman fan
  8. The permutahedral fan
  9. Tautological classes
  10. The $x$ and $y$ functions in the neighborhood of a cone of the permutahedral fan
  11. The Bergman fan and the Minkowski weights of the tautological functions
  12. Proof of Theorem \ref{['OmegaPosReduction']}
  13. Schubert matroids
  14. Relations in the group of polyhedra
  15. Relations in the group of matroid polytopes
  16. ...and 7 more sections

Key Result

Proposition 1.3

Let $M = M_1 \oplus M_2$. Then $g_M(t) = g_{M_1}(t) g_{M_2}(t)$ and $\omega(M) = \omega(M_1) \omega(M_2)$.

Figures (2)

  • Figure 1: The Ferroni paths and lattice points $\operatorname{verts}(S_{\bullet}, a_{\bullet})$ from Example \ref{['FerroniEx']}.
  • Figure 2: Illustration of Proposition \ref{['StressAndFerroni']}. The Ferroni paths for a rank $r$ matroid on $n$ elements make up the grid of dashed lines. The crowding coordinate in the picture is $x-y$: that is, sets $S$ with $p_M(S)=(x,y)$ have $\mathop{\mathrm{crowd}}\nolimits(S)=x-y$. The lines on which sets of crowding $0$ and $n-2r$ lie have been drawn in.

Theorems & Definitions (120)

  • Conjecture 1.1
  • Conjecture 1.2
  • Proposition 1.3
  • proof
  • Conjecture 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 2.1
  • Lemma 2.2
  • ...and 110 more