The $\mathcal{G}$-invariant and catenary data of a matroid

Abstract

The catenary data of a matroid $M$ of rank $r$ on $n$ elements is the vector $(ν(M;a_0,a_1,\ldots,a_r))$, indexed by compositions $(a_0,a_1,\ldots,a_r)$, where $a_0 \geq 0$,\, $a_i > 0$ for $i \geq 1$, and $a_0+ a_1 + \cdots + a_r = n$, with the coordinate $ν(M;a_0,a_1, \ldots,a_r)$ equal to the number of maximal chains or flags $(X_0,X_1, \ldots,X_r)$ of flats or closed sets such that $X_i$ has rank $i$,\, $|X_0| = a_0$, and $|X_i - X_{i-1}| = a_i$. We show that the catenary data of $M$ contains the same information about $M$ as its $\mathcal{G}$-invariant, which was defined by H. Derksen [\emph{J.\ Algebr.\ Combin.}\ 30 (2009) 43--86]. The Tutte polynomial is a specialization of the $\mathcal{G}$-invariant. We show that many known results for the Tutte polynomial have analogs for the $\mathcal{G}$-invariant. In particular, we show that for many matroid constructions, the $\mathcal{G}$-invariant of the construction can be calculated from the $\mathcal{G}$-invariants of the constituents and that the $\mathcal{G}$-invariant of a matroid can be calculated from its size, the isomorphism class of the lattice of cyclic flats with lattice elements labeled by the rank and size of the underlying set. We also show that the number of flats and cyclic flats of a given rank and size can be derived from the $\mathcal{G}$-invariant, that the $\mathcal{G}$-invariant of $M$ is reconstructible from the deck of $\mathcal{G}$-invariants of restrictions of $M$ to its copoints, and that, apart from free extensions and coextensions, one can detect whether a matroid is a free product from its $\mathcal{G}$-invariant.