Monotone Paths on Cross-Polytopes
Abstract
In the early 1990's, Billera and Sturmfels introduced the monotone path polytope (MPP), a special case of the general theory of fiber polytopes that associates a polytope to a pair (P,) of a polytope P and linear functional . In that same paper, they showed that MPPs of simplices and hyper-cubes are combinatorial cubes and permutahedra respectively. Their work has lead to many developments in combinatorics. Here we investigate the monotone paths for generic orientations of cross-polytopes. We show the face lattice of its MPP is isomorphic to the lattice of intervals in the sign poset from oriented matroid theory. We look at its f-vector, its realizations, and facets.
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