On exchange properties for Coxeter matroids and oriented matroids

We introduce new basis exchange axioms for matroids and oriented matroids. These new axioms are special cases of exchange properties for a more general class of combinatorial structures, Coxeter matroids. We refer to them as “properties” in the more general setting because they are not all equivalent, as they are for ordinary matroids, since the Symmetric Exchange Property is strictly stronger than the others. The weaker ones constitute the definition of Coxeter matroids, and we also prove their equivalence to the matroid polytope property of Gelfand and Serganova.