On Whitney duals of geometric lattices. (2023)
The concept of Whitney duality was first introduced by Gonz\'al\'ez D'Le\'on and Hallam in \cite{GONZALEZDLEON2021105301}. Two graded posets are said to be Whitney duals if they have their Whitney numbers of the first and second kind interchanged modulo sign. This is an interesting property being the Whitney numbers of a graded poset an important invariant in poset theory with connections to other mathematical contexts. The Whitney numbers appear, for example, as coefficients of chromatic polynomials of finite graphs. In \cite{GONZALEZDLEON2021105301} the authors also gave an explicit construction for Whitney duals under certain conditions, through the technique of EW-labelings. Some edge labelings that already appeared in the literature were shown to be EW-labelings, one particular case being the minimal edge labelings of geometric lattices introduced by Stanley. In this work we study specifically the Whitney duals of geometric lattices that arise from minimal EW-labelings. Since geometric lattices are in bijective correspondence to finite simple matroids, we aim to understand the construction of the Whitney duals only in terms of the information contained in an ordered matroid $(M,ω)$, where $M$ is a simple matroid and $ω$ is a total ordering of the ground set. To an ordered matroid one can associate its non-broken circuit complex (or NBC complex) as it was introduced by Bj\"orner in \cite{björner_1992}. We show that the Whitney dual corresponding to a minimal labeling of a geometric lattice can be described as a particular subposet of the NBC complex of its associated ordered matroid. More precisely, the subposet of the NBC complex formed by all the NBC sets and whose cover relations are determined by the removal of internally active elements on an NBC set is a Whitney dual to the lattice of flats of a matroid. % More specifically, it begins by giving a brief summary of necessary concepts to understand matroids and geometric lattices, as given in \cite{oxley2011matroid} and \cite{stanley2000enumerative}. Then, through the concept of NBC complexes given by Björner in \cite{björner_1992}, it proves that Whitney duals of geometric lattices formed through a special kind of Whitney labeling called minimal labeling can be described in terms of their corresponding matroid. Using this description we implement an algorithm using \href{http://www.sagemath.org/}{Sagemath} to construct the Whitney dual of the lattice of flats of an ordered matroid. We use this implementation to prove computationally that Whitney duals from different minimal labelings of a geometric lattice are not necessarily isomorphic. We compute the specific isomorphism classes (which here we refer as to atom ordering classes) of Whitney duals corresponding to minimal labelings for particular examples of matroids. In particular we determine that the Fano matroid has $5$ atom ordering classes and the non-Fano matroid has $42$ such classes.