Combinatorial Geometry in Characteristic 1

Many geometries over fields have formal analogues which can be thought of as geometries over the field of 1 element1. For example, the projective plane over the field Fq has q2 + q+ 1 points and the same number of lines; every line in the plane has q + 1 points. When q = 1, we have a plane with three points and three lines, i.e. a triangle. The flag complex of the triangle is a thin building of type A2 = Sym3. In general, the Coxeter complex W of the Coxeter group W is a thin building of type W and behaves like the building of type W over the field of 1 element. In this work we define the combinatorial flag varieties Ωn, n = 2, 3, . . . , which behave in many aspects as buildings of type An−1 over the field of 1 element but have the much more rich structure than Coxeter complexes. Their most surprising property is that every matroid on n elements is representable in Ωn. We start with three equivalent definitions of Ωn.