The signed circuits of an oriented matroid have indicator vectors whose convex hull is the oriented matroid circuit (OMC) polytope. This construction turns the combinatorics of signed circuits into a centrally symmetric polytope in which every circuit is a vertex, and its dimension and face structure encode properties of the underlying (graphical or cographical) matroid. The type-A family, built from the positive roots of the A_n root system, coincides with a graphic zonotope and with the polar dual of a symmetric edge polytope, linking it to Ehrhart theory and, through a symmetric-group action, to equivariant Ehrhart theory. Students learn how to read polytope structure, dimension, and lattice-point arithmetic off the combinatorics of an oriented matroid.
Given an oriented matroid, its signed circuits have indicator vectors with entries in {-1,0,+1}; the convex hull of these vectors is the oriented matroid circuit (OMC) polytope studied here. The pape…