Fiber polytopes for the projections between cyclic polytopes

The cyclic polytope $C(n,d)$ is the convex hull of any $n$ points on the moment curve ${(t,t^2,...,t^d):t \in \reals}$ in $\reals^d$. For $d' >d$, we consider the fiber polytope (in the sense of Billera and Sturmfels) associated to the natural projection of cyclic polytopes $\pi: C(n,d') \to C(n,d)$ which "forgets" the last $d'-d$ coordinates. It is known that this fiber polytope has face lattice indexed by the coherent polytopal subdivisions of $C(n,d)$ which are induced by the map $\pi$. Our main result characterizes the triples $(n,d,d')$ for which the fiber polytope is canonical in either of the following two senses: - all polytopal subdivisions induced by $\pi$ are coherent, - the structure of the fiber polytope does not depend upon the choice of points on the moment curve. We also discuss a new instance with a positive answer to the Generalized Baues Problem, namely that of a projection $\pi:P\to Q$ where $Q$ has only regular subdivisions and $P$ has two more vertices than its dimension.Comment: 28 pages with 1 postscript figur

Pseudodeterminants and perfect square spanning tree counts

The pseudodeterminant $\textrm{pdet}(M)$ of a square matrix is the last nonzero coefficient in its characteristic polynomial; for a nonsingular matrix, this is just the determinant. If $\partial$ is a symmetric or skew-symmetric matrix then $\textrm{pdet}(\partial\partial^t)=\textrm{pdet}(\partial)^2$. Whenever $\partial$ is the $k^{th}$ boundary map of a self-dual CW-complex $X$, this linear-algebraic identity implies that the torsion-weighted generating function for cellular $k$-trees in $X$ is a perfect square. In the case that $X$ is an \emph{antipodally} self-dual CW-sphere of odd dimension, the pseudodeterminant of its $k$th cellular boundary map can be interpreted directly as a torsion-weighted generating function both for $k$-trees and for $(k-1)$-trees, complementing the analogous result for even-dimensional spheres given by the second author. The argument relies on the topological fact that any self-dual even-dimensional CW-ball can be oriented so that its middle boundary map is skew-symmetric.Comment: Final version; minor revisions. To appear in Journal of Combinatoric

Diameter and Coherence of Monotone Path Graph

University of Minnesota Ph.D. dissertation. May 2015. Major: Mathematics. Advisor: Victor Reiner. 1 computer file (PDF); ix, 93 pages.A Zonotope $Z$ is the linear projection of an $n$-cube into $\mathbb{R}^d$. Given a generic linear function $f$, an $f$-monotone path on $Z$ is a path along edges from the $f$-minimizing vertex $-z$ to its opposite vertex $z$. The monotone paths of $Z$ are the vertices of the monotone path graph in which two $f$-monotone paths are adjacent when they differ in a face of $Z$. In our illustration the two red paths are adjacent in the monotone path graph because they differ in the highlighted face. An $f$-monotone path is coherent if it lies on the boundary of a polygon obtained by projecting $Z$ to 2 dimensions. The dotted, thick, red path in Figure 0.1 (see pdf) is coherent because it lies on the boundary after projecting $Z$ to the page. However, there is no equivalent projection for the blue double path. The alternate red path may be coherent or incoherent based on the choice of $f$. The coherent $f$-monotone paths of $Z$ are a set of geometrically distinguished galleries of the monotone path graph. Classifying when incoherent $f$-monotone paths exist is the central question of this thesis. We provide a complete classification of all monotone path graphs in corank 1 and 2, finding all families in which every $f$-monotone path is coherent and showing that all other zonotopes contain at least one incoherent $f$-monotone path. For arrangements of corank 1, we prove that the monotone path graph has diameter equal to the lower bound suggested by Reiner and Roichman using methods of $L_2$-accessibility and illustrate that $L_2$ methods cannot work in corank 2 by finding a monotone path graph which has no $L_2$-accessible nodes. We provide examples to illustrate the monotone path graph and obtain a variety of computational results, of which some are new while others confirm results obtained through different methods. Our primary methods use duality to reformulate coherence as a system of linear inequalities. We classify monotone path graphs using single element liftings and extensions, proving for when $Z$ has incoherent $f$-monotone paths, then any lifting or extension of $Z$ has incoherent $f$-monotone paths too. We complete our classification by finding all monotone path graphs with only coherent $f$-monotone paths and finding a set of minimal obstructions which always have incoherent $f$-monotone paths