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The $cd$-index of base polytopes for connected split matroids

Tommaso Faustini, Alejandro Vargas

TL;DR

This work derives a computable formula for the cd-index Ψ_cd of matroid base polytopes for connected split matroids by exploiting hyperplane-split decompositions and the cyclic-flat structure. The main result expresses Ψ_cd(M) in terms of the hypersimplex cd-index, plus corrections from cyclic flats and modular-pair interactions, with an explicit, data-minimal set of matroidal parameters λ(r,h) and μ(α,β;a,b). The authors develop and demonstrate the method on cuspidal and sparse paving matroids, provide a general recursion, and supply code to perform computations, thereby enabling efficient cd-index enumeration beyond previously tractable cases. They also pose open questions about extending these techniques to other 0/1-polytopes and arbitrary matroids, and about deeper structural inequalities governing the flag data of matroid polytopes.

Abstract

We compute the $cd$-index $Ψ_{cd}$ of matroid base polytopes $\mathscr{P}(M)$ for a large family of matroids $M$. The $cd$-index is a polynomial in two non-commutative variables that compactly encodes the count of face flags $\mathcal{F} = \{σ_1 \subset \dots \subset σ_s \}$ with prescribed $\dim σ_i = d_i$. This comprises the $f$-vector of $\mathscr{P}(M)$, which recently Ferroni and Schröter treated as an almost-valuative invariant; i.e. a valuative part plus an error term. We initiate a similar program for $Ψ_{cd}(\mathscr{P}(M))$ and show that for an elementary split matroid $M$ the error term in the computation of $Ψ_{cd}(\mathscr{P}(M))$ surprisingly depends only on modular pairs of cyclic flats. This allows us to implement computations requiring only the counts $λ(r,h)$ and $μ(α,β,a,b)$ of cyclic flats and modular pairs of cyclic flats, respectively, that fulfill some rank and cardinality conditions. We illustrate the methods with sparse paving matroids.

The $cd$-index of base polytopes for connected split matroids

TL;DR

This work derives a computable formula for the cd-index Ψ_cd of matroid base polytopes for connected split matroids by exploiting hyperplane-split decompositions and the cyclic-flat structure. The main result expresses Ψ_cd(M) in terms of the hypersimplex cd-index, plus corrections from cyclic flats and modular-pair interactions, with an explicit, data-minimal set of matroidal parameters λ(r,h) and μ(α,β;a,b). The authors develop and demonstrate the method on cuspidal and sparse paving matroids, provide a general recursion, and supply code to perform computations, thereby enabling efficient cd-index enumeration beyond previously tractable cases. They also pose open questions about extending these techniques to other 0/1-polytopes and arbitrary matroids, and about deeper structural inequalities governing the flag data of matroid polytopes.

Abstract

We compute the -index of matroid base polytopes for a large family of matroids . The -index is a polynomial in two non-commutative variables that compactly encodes the count of face flags with prescribed . This comprises the -vector of , which recently Ferroni and Schröter treated as an almost-valuative invariant; i.e. a valuative part plus an error term. We initiate a similar program for and show that for an elementary split matroid the error term in the computation of surprisingly depends only on modular pairs of cyclic flats. This allows us to implement computations requiring only the counts and of cyclic flats and modular pairs of cyclic flats, respectively, that fulfill some rank and cardinality conditions. We illustrate the methods with sparse paving matroids.

Paper Structure

This paper contains 19 sections, 16 theorems, 43 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Counting chains of faces
  3. Flag $f$-vectors and the cd-index
  4. Cuts and products
  5. A mixed expression
  6. The cd-index of the hypersimplex $\Delta_{k,n}$
  7. Matroids and one-cuts from $\Delta_{k,n}$
  8. Matroids as slices of $\Delta_{k,n}$
  9. Connected split matroids
  10. Cuspidal matroids
  11. The cd-index of connected split matroids
  12. The case with two cyclic flats
  13. The general formula
  14. Sparse paving matroids
  15. Code for computation
  16. ...and 4 more sections

Key Result

Proposition 2.1

Let $\Pi$ be a finite poset. There exists an integer-coefficient polynomial $\Psi_{cd}(\Pi)$ such that the substitution $c = a + b$ and $d = ab + ba$ yields $\Psi_{ab}(\Pi)$ if and only if $\Pi$ satisfies the generalized Dehn-Sommerville relations.

Figures (2)

  • Figure 1: Two hyperplanes $H_F$ and $H_G$ cut the space into four regions labeled $++, -+, --, +-$. A face of $\Delta_{k,n}$ like the bipyramid $\sigma$ is possible, whereas a face like the square is not since a piece lies in the $++$ region. If $\hat{\sigma} = \sigma \cap H_G$, then Corollary \ref{['cor:FaceOnlySplitInTwo']} says that $\sigma$ lives entirely in the $-+$ and $+-$ regions if and only if $\hat{\sigma}$ is in the intersection $H_F \cap H_G$
  • Figure 2: Slice of $\Delta_{3,5}$ with two vertices deleted, i.e. 11100 and 10011. Not all the edges between vertices are shown. We show a pair of deletion and contraction facets (simplex and bipyramid), together with the intersection of two cuts $H_F = H(123,2)$ and $H_G = H(145,2)$ corresponding to a modular pair of flats (the square $\tau$ in the middle). The face $\sigma \in \Delta_{k,n}$ for which $\tau = \sigma \cap H_G$ equals the bipyramid made with the deleted vertices. They live in the regions $-+$ and $+-$, illustrating Corollary \ref{['cor:FaceOnlySplitInTwo']}.

Theorems & Definitions (31)

  • Proposition 2.1: Thm 4 of bk91
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • Remark 2.7
  • ...and 21 more