UC Berkeley

Extrapolating from the work of Las Vergnas on the external active order for matroid bases, and inspired by the structure of Lenz's forward exchange matroids in the theory of zonotopal algebra, we develop the combinatorial theory of the generalized external order. This partial ordering on the independent sets of an ordered matroid is a supersolvable join-distributive lattice which is a refinement of the geometric lattice of flats, and is fundamentally derived from the classical notion of matroid activity. We uniquely classify the lattices which occur as the external order of an ordered matroid, and we explore the intricate structure of the lattice's downward covering relations, as well as its behavior under deletion and contraction of the underlying matroid.

We then apply this theory to improve our understanding of certain constructions in zonotopal algebra. We first explain the fundamental link between zonotopal algebra and the external order by characterizing Lenz's forward exchange matroids in terms of the external order. Next we describe the behavior of Lenz's zonotopal $\mathcal{D}$-basis polynomials under taking directional derivatives, and we use this understanding to provide a new algebraic construction for these polynomials. The construction in particular provides the first known algorithm for computing these polynomials which is computationally tractible for inputs of moderate size. Finally, we provide an explicit construction for the zonotopal $\mathcal{P}$-basis polynomials for the internal and semi-internal settings.