Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Shellability of the quotient order on lattice path matroids

Carolina Benedetti, Anton Dochtermann, Kolja Knauer, Yupeng Li

TL;DR

The paper addresses the topology of the quotient poset ${\mathcal{P}}_n$ of lattice path matroids by establishing an EL-labeling via a good pair labeling, which yields shellability of the order complex and a combinatorial handle on the Möbius function through falling chains encoded by permutation data. It connects the combinatorics of LPM quotients to permutations, offering an explicit Bruhat-order perspective and proving monotonicity results and exact counts in key cases. The authors further show that the labeling satisfies Whitney-criteria, producing a Whitney dual poset $Q_{\lambda}({\mathcal{P}}_n)$ and illustrating it with a concrete ${\mathcal{P}}_3$ example. Overall, the work provides structural, topological, and enumerative insights into the quotient poset of LPMs and opens questions about broader connections to positroid flags and nonnegative flag varieties.

Abstract

The concept of a matroid quotient has connections to fundamental questions in the geometry of flag varieties. In previous work, Benedetti and Knauer characterized quotients in the class of lattice path matroids (LPMs) in terms of a simple combinatorial condition. As a consequence, they showed that the quotient order on LPMs yields a graded poset whose rank polynomial relates to a refinement of the Catalan numbers. In this work we show that this poset admits an EL-labeling, implying that the order complex is shellable and hence enjoys several combinatorial and topological properties. We use this to establish bounds on the Möbius function of the poset, interpreting falling chains in the EL-labeling in terms of properties of underlying permutations. Furthermore, we show that this EL-labeling is in fact a Whitney labeling, in the sense of the recent notion introduced by González D'León and Hallam.

Shellability of the quotient order on lattice path matroids

TL;DR

The paper addresses the topology of the quotient poset of lattice path matroids by establishing an EL-labeling via a good pair labeling, which yields shellability of the order complex and a combinatorial handle on the Möbius function through falling chains encoded by permutation data. It connects the combinatorics of LPM quotients to permutations, offering an explicit Bruhat-order perspective and proving monotonicity results and exact counts in key cases. The authors further show that the labeling satisfies Whitney-criteria, producing a Whitney dual poset and illustrating it with a concrete example. Overall, the work provides structural, topological, and enumerative insights into the quotient poset of LPMs and opens questions about broader connections to positroid flags and nonnegative flag varieties.

Abstract

The concept of a matroid quotient has connections to fundamental questions in the geometry of flag varieties. In previous work, Benedetti and Knauer characterized quotients in the class of lattice path matroids (LPMs) in terms of a simple combinatorial condition. As a consequence, they showed that the quotient order on LPMs yields a graded poset whose rank polynomial relates to a refinement of the Catalan numbers. In this work we show that this poset admits an EL-labeling, implying that the order complex is shellable and hence enjoys several combinatorial and topological properties. We use this to establish bounds on the Möbius function of the poset, interpreting falling chains in the EL-labeling in terms of properties of underlying permutations. Furthermore, we show that this EL-labeling is in fact a Whitney labeling, in the sense of the recent notion introduced by González D'León and Hallam.

Paper Structure

This paper contains 11 sections, 16 theorems, 22 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Preliminaries
  4. Matroids and quotients
  5. Shelling and EL-labelings
  6. Shellings and and Möbius functions of P
  7. Möbius functions
  8. Counting falling chains
  9. Whitney duality
  10. Further questions
  11. Acknowledgments

Key Result

Theorem 3

BenKna Let $M=M[U,L]$ and $M'=M[U',L']$ be LPMs on the ground set $[n]$. We have that $M' \leq_q M$ if and only if $U'\subseteq U$, $L'\subseteq L$ and the greedy pairing of $(L\setminus L',U\setminus U')$ is good.

Figures (5)

  • Figure 1: A basis in the diagram representing the LPM $M[1246,3568]$.
  • Figure 2: An LPM with a bad pair $(\ell_i, u_j)$. Exactly those upward segments $\ell\in L$ on the dotted path yield good pairs with $u_j$.
  • Figure 3: The EL-labeling of ${\mathcal{P}_3}$, along with six falling chains.
  • Figure 4: Left: the 17 maximal chains of the poset $P_3$. Right: The Whitney equivalence classes partitioned into 6 graphs. Blue labels on edges correspond to equivalences in $C_3$.
  • Figure 5: Top: $C_3$. Bottom: Whitney dual ${\mathcal{Q}}_3$ of ${\mathcal{P}}_3$. To reduce notation, edges with the same label are drawn with the same slope.

Theorems & Definitions (39)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Lemma 4
  • Definition 5
  • Theorem 6
  • Theorem 7
  • proof
  • Remark 8
  • Definition 9
  • ...and 29 more