We introduce the notion of real phase structure on rational polyhedral fans in Euclidean space. Such a structure consists of an assignment of affine spaces over $mathbb{Z}/2mathbb{Z}$ to each top dimensional face of the fan subject to two conditions. Given an oriented matroid we can construct a real phase structure on the fan of the underlying matroid. Conversely, we show that from a real phase structure on a matroid fan we can produce an orientation of the underlying matroid. Thus real phase structures are cryptomorphic to matroid orientations. The topes of the orientated matroid are recovered immediately from the real phase structure. We also provide a direct way to recover the signed circuits of the oriented matroid from the real phase structure.
We express the matroid polytope $P_M$ of a matroid $M$ as a signed Minkowski sum of simplices, and obtain a formula for the volume of $P_M$. This gives a combinatorial expression for the degree of an arbitrary torus orbit closure in the Grassmannian $Gr_{k,n}$. We then derive analogous results for the independent set polytope and the associated flag matroid polytope of $M$. Our proofs are based on a natural extension of Postnikovs theory of generalized permutohedra.
We give a criterion for modular extension of rank-4 hypermodular matroids, and prove a weakening of Kantors conjecture for rank-4 realizable matroids. This proves the sticky matroid conjecture and Kantors conjecture for realizable matroids due to an argument of Bachem, Kern, and Bonin, and due to an equivalence argument of Hochstattler and Wilhelmi, respectively.
We show Kantors conjecture (1974) holds in rank 4. This proves both the sticky matroid conjecture of Poljak and Turzik (1982) and the whole Kantors conjecture, due to an argument of Bachem, Kern, and Bonin, and an equivalence argument of Hochstattler and Wilhelmi, respectively.
We endow the set of isomorphic classes of matroids with a new Hopf algebra structure, in which the coproduct is implemented via the combinatorial operations of restriction and deletion. We also initiate the investigation of dendriform coalgebra structures on matroids and introduce a monomial invariant which satisfy a convolution identity with respect to restriction and deletion.
Motivated by a rigidity-theoretic perspective on the Localization Problem in 2D, we develop an algorithm for computing circuit polynomials in the algebraic rigidity matroid associated to the Cayley-Menger ideal for $n$ points in 2D. We introduce combinatorial resultants, a new operation on graphs that captures properties of the Sylvester resultant of two polynomials in the algebraic rigidity matroid. We show that every rigidity circuit has a construction tree from $K_4$ graphs based on this operation. Our algorithm performs an algebraic elimination guided by the construction tree, and uses classical resultants, factorization and ideal membership. To demonstrate its effectiveness, we implemented our algorithm in Mathematica: it took less than 15 seconds on an example where a Groebner Basis calculation took 5 days and 6 hrs.