The Polyhedral Geometry of Partially Ordered Sets

Pairs of polyhedra connected by a piecewise-linear bijection appear in different fields of mathematics. The model example of this situation are the order and chain polytopes introduced by Stanley in, whose defining inequalities are given by a finite partially ordered set. The two polytopes have different face lattices, but admit a volume and lattice point preserving piecewise-linear bijection called the transfer map. Other areas like representation theory and enumerative combinatorics provide more examples of pairs of polyhedra that are similar to order and chain polytopes. The goal of this thesis is to analyze this phenomenon and move towards a common theoretical framework describing these polyhedra and their piecewise-linear bijections. A first step in this direction was done by Ardila, Bliem and Salazar, where the authors generalize order and chain polytopes by replacing the defining data with a marked poset. These marked order and chain polytopes still admit a piecewise-linear transfer map and include the Gelfand-Tsetlin and Feigin-Fourier-Littelmann-Vinberg polytopes from representation theory among other examples. We consider more polyhedra associated to marked posets and obtain new results on their face structure and combinatorial interplay. Other examples found in the literature bear resemblance to these marked poset polyhedra but do not admit a description as such. This is our motivation to consider distributive polyhedra, which are characterized by describing networks by Felsner and Knauer analogous to the description of order polytopes by Hasse diagrams. For a subclass of distributive polyhedra we are able to construct a piecewise-linear bijection to another polyhedron related to chain polytopes. We give a description of this transfer map and the defining inequalities of the image in terms of the underlying network