Sweeps, polytopes, oriented matroids, and allowable graphs of permutations

A sweep of a point configuration is any ordered partition induced by a linear functional. Posets of sweeps of planar point configurations were formalized and abstracted by Goodman and Pollack under the theory of allowable sequences of permutations. We introduce two generalizations that model posets of sweeps of higher dimensional configurations. Mimicking the fact that sweep polytopes of point configurations (the monotone path polytopes of the associated zonotopes) are projections of permutahedra, we define sweep oriented matroids as strong maps of the braid oriented matroid. Allowable sequences are then the sweep oriented matroids of rank 2, and many of their properties extend to higher rank. We show strong ties between sweep oriented matroids and both modular hyperplanes and Dilworth truncations from (unoriented) matroid theory. We also explore their connection with the generalized Baues problem for cellular strings, where sweep oriented matroids can play the role of monotone path polytopes, even for non-realizable oriented matroids. In particular, we show that for oriented matroids that admit a sweep oriented matroid, their poset of pseudo-sweeps deformation retracts to a sphere of the appropriate dimension. A second generalization are allowable graphs of permutations: symmetric sets of permutations pairwise connected by allowable sequences. They have the structure of acycloids and include sweep oriented matroids.