When the standard representation of a crystallographic Coxeter group $Γ$ is reduced modulo an odd prime $p$, a finite representation in some orthogonal space over $\mathbb{Z}_p$ is obtained. If $Γ$ has a string diagram, the latter group will often be the automorphism group of a finite regular polytope. In Part I we described the basics of this construction and enumerated the polytopes associated with the groups of rank 3 and the groups of spherical or Euclidean type. In this paper, we investigate such families of polytopes for more general choices of $Γ$, including all groups of rank 4. In particular, we study in depth the interplay between their geometric properties and the algebraic structure of the corresponding finite orthogonal group.
When the standard representation of a crystallographic Coxeter group is reduced modulo an odd prime p, one obtains a finite group G^p acting on some orthogonal space over Z_p . If the Coxeter group has a string diagram, then G^p will often be the automorphism group of a finite abstract regular polytope. In parts I and II we established the basics of this construction and enumerated the polytopes associated to groups of rank at most 4, as well as all groups of spherical or Euclidean type. Here we extend the range of our earlier criteria for the polytopality of G^p . Building on this we investig…
The $f$-vector of a polytope consists of the numbers of its $i$-dimensional faces. An open field of study is the characterization of all possible $f$-vectors. It has been solved in three dimensions by Steinitz in the early 19th century. We state a related question, i.e. to characterize $f$-vectors of three dimensional polytopes respecting a symmetry, given by a finite group of matrices. We give a full answer for all three dimensional polytopes that are symmetric with respect to a finite rotation or rotary reflection group. We solve these cases constructively by developing tools that generalize…
We show that the Grothendieck group associated to integral polytopes in $\mathbb{R}^n$ is free-abelian by providing an explicit basis. Moreover, we identify the involution on this polytope group given by reflection about the origin as a sum of Euler characteristic type. We also compute the kernel of the norm map sending a polytope to its induced seminorm on the dual of $\mathbb{R}^n$.
Let $P$ be a convex polytope in the Euclidean space $\E^n$. Consider the group $G_P$ generated by reflections in the facets of $P$. We say that $P$ {\it generates a reflection group $G_P$}. In the present paper, we list all Euclidean simplices generating discrete reflection groups.
Let $G$ be a discrete group generated by reflections in hyperbolic or Euclidean space, and $H\subset G$ be a finite index subgroup generated by reflections. Suppose that the fundamental chamber of $G$ is a finite volume polytope with $k$ facets. In this paper, we prove that the fundamental chamber of $H$ has at least $k$ facets.
In the classical setting, a convex polytope is said to be semiregular if its facets are regular and its symmetry group is transitive on vertices. This paper studies semiregular abstract polytopes, which have abstract regular facets, still with combinatorial automorphism group transitive on vertices. We analyze the structure of the automorphism group, focusing in particular on polytopes with two kinds of regular facets occurring in an "alternating" fashion. In particular we use group amalgamations to prove that given two compatible n-polytopes P and Q, there exists a universal abstract semiregu…
We describe the canonical correspondence between set of all finite metric spaces and set of special symmetric convex polytopes, and formulate the problem about classification of the metric spaces in terms of combinatorial structure of those polytopes.
We prove that every finite group is the automorphism group of a finite abstract polytope isomorphic to a face-to-face tessellation of a sphere by topological copies of convex polytopes. We also show that this abstract polytope may be realized as a convex polytope.
Points of an orbit of a finite Coxeter group G, generated by n reflections starting from a single seed point, are considered as vertices of a polytope (G-polytope) centered at the origin of a real n-dimensional Euclidean space. A general efficient method is recalled for the geometric description of G- polytopes, their faces of all dimensions and their adjacencies. Products and symmetrized powers of G-polytopes are introduced and their decomposition into the sums of G-polytopes is described. Several invariants of G-polytopes are found, namely the analogs of Dynkin indices of degrees 2 and 4, an…
An orbit polytope is the convex hull of an orbit under a finite group $G \leq \operatorname{GL}(d,\mathbb{R})$. We develop a general theory of possible affine symmetry groups of orbit polytopes. For every group, we define an open and dense set of generic points such that the orbit polytopes of generic points have conjugated affine symmetry groups. We prove that the symmetry group of a generic orbit polytope is again $G$ if $G$ is itself the affine symmetry group of some orbit polytope, or if $G$ is absolutely irreducible. On the other hand, we describe some general cases where the affine symme…
Hilbert space frames generalize orthonormal bases to allow redundancy in representations of vectors while keeping good reconstruction properties. A frame comes with an associated frame operator encoding essential properties of the frame. We study a polytope that arises in an algorithm for constructing all finite frames with given lengths of frame vectors and spectrum of the frame operator, which is a Gelfand-Tsetlin polytope. For equal norm tight frames, we give a non-redundant description of the polytope in terms of equations and inequalities. From this we obtain the dimension and number of f…
The "edge polytope" of a finite graph G is the convex hull of the columns of its vertex-edge incidence matrix. We study extremal problems for this class of polytopes. For k =2, 3, 5 we determine the maximum number of vertices of k-neighborly edge polytopes up to a sublinear term. We also construct a family of edge polytopes with exponentially-many facets.
For each strongly connected finite-dimensional (pure) simplicial complex we construct a finite group, the group of projectivities of the complex, which is a combinatorial but not a topological invariant. This group is studied for combinatorial manifolds and, in particular, for polytopal simplicial spheres. The results are applied to a coloring problem for simplicial (or, dually, simple) polytopes which arises in the area of toric algebraic varieties.
Any convex polytope whose combinatorial automorphism group has two orbits on the flags is isomorphic to one whose group of Euclidean symmetries has two orbits on the flags (equivalently, to one whose automorphism group and symmetry group coincide.) Hence, a combinatorially two-orbit convex polytope is isomorphic to one of a known finite list, all of which are 3-dimensional: the cuboctahedron, icosidodecahedron, rhombic dodecahedron, or rhombic triacontahedron. The same is true of combinatorially two-orbit normal face-to-face tilings by convex polytopes.
A split of a polytope $P$ is a (regular) subdivision with exactly two maximal cells. It turns out that each weight function on the vertices of $P$ admits a unique decomposition as a linear combination of weight functions corresponding to the splits of $P$ (with a split prime remainder). This generalizes a result of Bandelt and Dress [Adv. Math. 92 (1992)] on the decomposition of finite metric spaces. Introducing the concept of compatibility of splits gives rise to a finite simplicial complex associated with any polytope $P$, the split complex of $P$. Complete descriptions of the split complexe…
The paper studies modular reduction techniques for abstract regular and chiral polytopes, with two purposes in mind: first, to survey the literature about modular reduction in polytopes; and second, to apply modular reduction, with moduli given by primes in Z[t] (with t=τthe golden ratio), to construct new regular 4-polytopes of hyperbolic types {3,5,3} and {5,3,5} with automorphism groups given by finite orthogonal groups.
There are two main thrusts in the theory of regular and chiral polytopes: the abstract, purely combinatorial aspect, and the geometric one of realizations. This brief survey concentrates on the latter. The dimension of a faithful realization of a finite abstract regular polytope in some euclidean space is no smaller than its rank, while that of a chiral polytope must strictly exceed the rank. There are similar restrictions on the dimensions of realizations of regular and chiral apeirotopes. From the viewpoint of realizations in a fixed dimension, the problems are now completely solved in up to…
We study finite groups that occur as combinatorial automorphism groups or geometric symmetry groups of convex polytopes. When $Γ$ is a subgroup of the combinatorial automorphism group of a convex $d$-polytope, $d\geq 3$, then there exists a convex $d$-polytope related to the original polytope with combinatorial automorphism group exactly $Γ$. When $Γ$ is a subgroup of the geometric symmetry group of a convex $d$-polytope, $d\geq 3$, then there exists a convex $d$-polytope related to the original polytope with both geometric symmetry group and combinatorial automorphism group exactly $Γ$. These…
When the standard representation of a crystallographic Coxeter group G (with string diagram) is reduced modulo the integer d>1, one obtains a finite group G^d which is often the automorphism group of an abstract regular polytope. Building on earlier work in the case that d is an odd prime, we here develop methods to handle composite moduli and completely describe the corresponding modular polytopes when G is of spherical or Euclidean type. Using a modular variant of the quotient criterion, we then describe the locally toroidal polytopes provided by our construction, most of which are new.
We classify the convex polytopes whose symmetry groups have two orbits on the flags. These exist only in two or three dimensions, and the only ones whose combinatorial automorphism group is also two-orbit are the cuboctahedron, the icosidodecahedron, and their duals. The combinatorially regular two-orbit convex polytopes are certain 2n-gons for each n > 1. We also classify the face-to-face tilings of Euclidean space by convex polytopes whose symmetry groups have two flag orbits. There are finitely many families, tiling one, two, or three dimensions. The only such tilings which are also combina…
In this paper, we classify all the hyperbolic non-compact Coxeter polytopes of finite volume combinatorial type of which is either a pyramid over a product of two simplices or a product of two simplices of dimension greater than one. Combined with results of Kaplinskaja (1974) and Esselmann (1996) this completes the classification of hyperbolic Coxeter n-polytopes of finite volume with n+2 facets.
Starting from a finite simple graph $G$, for each eigenvalue $θ$ of its adjacency matrix one can construct a convex polytope $P_G(θ)$, the so called $θ$-eigenpolytop of $G$. For some polytopes this technique can be used to reconstruct the polytopes from its edge-graph. Such polytopes (we shall call them spectral) are still badly understood. We give an overview of the literature for eigenpolytopes and spectral polytopes. We introduce a geometric condition by which to prove that a given polytope is spectral (more exactly, $θ_2$-spectral). We apply this criterion to the edge-transitive polytopes.…
We define certain natural finite sums of $n$'th roots of unity, called $G_P(n)$, that are associated to each convex integer polytope $P$, and which generalize the classical $1$-dimensional Gauss sum $G(n)$ defined over $\mathbb Z/ {n \mathbb Z}$, to higher dimensional abelian groups and integer polytopes. We consider the finite Weyl group $\mathcal{W}$, generated by the reflections with respect to the coordinate hyperplanes, as well as all permutations of the coordinates; further, we let $\mathcal G$ be the group generated by $\mathcal{W}$ as well as all integer translations in $\mathbb Z^d$. …
A polynomial representation of a convex d-polytope P is a finite set \{p_1(x),...,p_n(x)\} of polynomials over E^d such that P=\setcond{x \in \E^d}{p_1(x) \ge 0 {for every} 1 \le i \le n}. By s(d,P) we denote the least possible number of polynomials in a polynomial representation of P. It is known that d \le s(d,P) \le 2d-1. Moreover, it is conjectured that s(d,P)=d for all convex d-polytopes P. We confirm this conjecture for simple d-polytopes by providing an explicit construction of d polynomials that represent a given simple d-polytope P.
We study Voronoi cells in the statistical setting by considering preimages of the maximum likelihood estimator that tessellate an open probability simplex. In general, logarithmic Voronoi cells are convex sets. However, for certain algebraic models, namely finite models, models with ML degree 1, linear models, and log-linear (or toric) models, we show that logarithmic Voronoi cells are polytopes. As a corollary, the algebraic moment map has polytopes for both its fibres and its image, when restricted to the simplex. We also compute non-polytopal logarithmic Voronoi cells using numerical algebr…
