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

TL;DR

This paper computes the $cd$-index of matroid base polytopes for a broad class of matroids, revealing that for elementary split matroids, the error term depends only on modular pairs of cyclic flats, enabling efficient calculations.

Contribution

It introduces a method to compute the $cd$-index for elementary split matroids, showing the error term depends solely on modular pairs of cyclic flats, simplifying calculations.

Findings

The $cd$-index can be computed using counts of cyclic flats and modular pairs.

For elementary split matroids, the error term depends only on modular pairs.

The methods are demonstrated on sparse paving matroids.

Abstract

We compute the $cd$-index $\Psi_{cd}$ 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 $\mathcal{F} = \{\sigma_1 \subset \dots \subset \sigma_s \}$ with prescribed $\dim \sigma_i = d_i$. This comprises the $f$-vector of $\mathscr{P}(M)$, which recently Ferroni and Schr\"oter treated as an almost-valuative invariant; i.e. a valuative part plus an error term. We initiate a similar program for $\Psi_{cd}(\mathscr{P}(M))$ and show that for an elementary split matroid $M$ the error term in the computation of $\Psi_{cd}(\mathscr{P}(M))$ surprisingly depends only on modular pairs of cyclic flats. This allows us to implement computations requiring only the counts $\lambda(r,h)$ and $\mu(\alpha,\beta,a,b)$ 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.