Geometry of Generalized Permutohedra
Year: 2011
Type: dissertation
Abstract: We study generalized permutohedra and some of the geometric properties they exhibit. We decompose matroid polytopes (and several related polytopes) into signed Minkowski sums of simplices and compute their volumes. We define the associahedron and multiplihedron in terms of trees and show them to be generalized permutohedra. We also generalize the multiplihedron to a broader class of generalized permutohedra, and describe their face lattices, vertices, and volumes. A family of interesting polynomials that we call composition polynomials arises from the study of multiplihedra, and we analyze several of their surprising properties. Finally, we look at generalized permutohedra of different root systems and study the Minkowski sums of faces of the crosspolytope.
Author: Jeffrey Doker
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2103742187