The Face Structure and Geometry of Marked Order Polyhedra

TL;DR

This paper explores the structure and geometry of marked order polyhedra, connecting combinatorial properties of posets with geometric features relevant in representation theory.

Contribution

It provides a combinatorial characterization of faces of marked order polyhedra and describes their geometric properties, including facets, recession cones, and Minkowski decompositions.

Findings

Faces correspond to partitions of the poset

Facets relate to covering relations in certain marked posets

Recession cones and Minkowski sums are characterized

Abstract

We study a class of polyhedra associated to marked posets. Examples of these polyhedra are Gelfand-Tsetlin polytopes and cones, as well as Berenstein-Zelevinsky polytopes, all of which have appeared in the representation theory of semi-simple Lie algebras. The faces of these polyhedra correspond to certain partitions of the underlying poset and we give a combinatorial characterization of these partitions. We specify a class of marked posets that give rise to polyhedra with facets in correspondence to the covering relations of the poset. On the convex geometrical side, we describe the recession cone of the polyhedra, discuss products and give a Minkowski sum decomposition. We briefly discuss intersections with affine subspaces that have also appeared in representation theory and recently in the theory of finite Hilbert space frames.