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Schubert matroids, Delannoy paths, and Speyer's invariant

Luis Ferroni

Abstract

We provide a combinatorial way of computing Speyer's $g$-polynomial on arbitrary Schubert matroids via the enumeration of certain Delannoy paths. We define a new statistic of a basis in a matroid, and express the $g$-polynomial of a Schubert matroid in terms of it and internal and external activities. Some surprising positivity properties of the $g$-polynomial of Schubert matroids are deduced from our expression. Finally, we combine our formulas with a fundamental result of Derksen and Fink to provide an algorithm for computing the $g$-polynomial of an arbitrary matroid.

Schubert matroids, Delannoy paths, and Speyer's invariant

Abstract

We provide a combinatorial way of computing Speyer's -polynomial on arbitrary Schubert matroids via the enumeration of certain Delannoy paths. We define a new statistic of a basis in a matroid, and express the -polynomial of a Schubert matroid in terms of it and internal and external activities. Some surprising positivity properties of the -polynomial of Schubert matroids are deduced from our expression. Finally, we combine our formulas with a fundamental result of Derksen and Fink to provide an algorithm for computing the -polynomial of an arbitrary matroid.
Paper Structure (13 sections, 14 theorems, 31 equations, 8 figures, 1 table, 1 algorithm)

This paper contains 13 sections, 14 theorems, 31 equations, 8 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Overview
  3. Main results
  4. Outline
  5. Preliminaries
  6. Schubert matroids
  7. Matroids and lattice paths
  8. The covaluative group
  9. Delannoy paths and Speyer's invariant
  10. Delannoy paths and Schubert matroids
  11. Subdivisions into direct sums of series-parallel matroids
  12. A reinterpretation of the Delannoy paths
  13. An algorithm for general matroids

Key Result

Theorem 1.4

Let $\mathsf{M}$ be a loopless and coloopless Schubert matroid. The $g$-polynomial of $\mathsf{M}$ is given by: where $c_i$ counts the number of admissible Delannoy paths associated to $\mathsf{M}$ having exactly $i-1$ diagonal steps.

Figures (8)

  • Figure 1: $n=12$, $r=5$ and $U=\{1,2,5,7,10\}$.
  • Figure 2: $B=\{3,4,5,7,12\}$.
  • Figure 3: Two non-examples (left) and two examples (right).
  • Figure 4: A Schubert matroid and the three snakes of the described subdivision.
  • Figure 5: Admissible Delannoy paths without diagonals.
  • ...and 3 more figures

Theorems & Definitions (35)

  • Conjecture 1.1: speyer-conjecture
  • Definition 1.2
  • Conjecture 1.3: speyer
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Definition 2.1
  • Proposition 2.2: bonin-demier
  • Example 2.3
  • ...and 25 more