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

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.