Dimitri Leemans, Egon Schulte
We prove that if G is a string C-group of rank 4 and G is isomorphic to L_2(q) with q a prime power, then q must be 11 or 19. The polytopes arising are Grunbaum's 11-cell of type {3,5,3} for L_2(11) and Coxeter's 57-cell of type {5,3,5} for L_2(19), each a locally projective regular 4-polytope.
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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.
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…
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…
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 paper gives a collection of open problems on abstract polytopes that were either presented at the Polytopes Day in Calgary or motivated by discussions at the preceding Workshop on Convex and Abstract Polytopes at the Banff International Research Station in May 2005.
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.
Given a Laurent polynomial f, one can form the period of f: this is a function of one complex variable that plays an important role in mirror symmetry for Fano manifolds. Mutations are a particular class of birational transformations acting on Laurent polynomials in two variables; they preserve the period and are closely connected with cluster algebras. We propose a higher-dimensional analog of mutation acting on Laurent polynomials f in n variables. In particular we give a combinatorial description of mutation acting on the Newton polytope P of f, and use this to establish many basic facts ab…
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.
Two-Orbit Polytopes
本文研究双轨道多面体，探讨其结构特性及自同构群，揭示其在不同类别中的行为差异。
A concept of generalized regular polytope is introduced in this work. The number of its (1...n-1)-dimensional elements is not necessarily integer, though all the combinatorial and metric properties meet those of regular polytopes in a classic sense. New relationships between Schlafli symbol of the regular polytope and its metric parameters have been established. Using the generalized regular polytopes concept, group and metric properties of arbitrary metric space tessellations into regular honeycombs were investigated. It has been shown that sequential tessellations of space into regular honey…
Inscribable fans I: Inscribed cones and virtual polytopes
本文研究内接于球面的多胞形，引入内接锥与内接虚拟多胞形，证明其封闭性及多项式时间可判定性，并分类二维可内接扇及若干多胞形类，关联反射群胚。
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…
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.
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 par…
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…
There exists just one regular polytope of rank larger than 3 whose full automorphism group is a projective general linear group PGL_2(q), for some prime-power q. This polytope is the 4-simplex and the corresponding group is PGL_2(5), which is isomorphic to S_5.
In this paper, we classify regular polytopes with automorphism groups of order $2^n$ and Schläfli types $\{4, 2^{n-3}\}, \{4, 2^{n-4}\}$ and $\{4, 2^{n-5}\}$ for $n \geq 10$, therefore giving a partial answer to a problem proposed by Schulte and Weiss in [Problems on polytopes, their groups, and realizations, Periodica Math. Hungarica 53(2006) 231-255].
Regular incidence complexes are combinatorial incidence structures generalizing regular convex polytopes, regular complex polytopes, various types of incidence geometries, and many other highly symmetric objects. The special case of abstract regular polytopes has been well-studied. The paper describes the combinatorial structure of a regular incidence complex in terms of a system of distinguished generating subgroups of its automorphism group or a flag-transitive subgroup. Then the groups admitting a flag-transitive action on an incidence complex are characterized as generalized string C-group…
Polytopes are ubiquitous in different areas of mathematics. Gleason and Hubard established a factorisation theorem, stating that every abstract polytope has a unique factorisation into prime polytopes. We compute the automorphism group of a certain family of inductively constructed polytopes using the unique factorisation.
We construct two infinite families of locally toroidal chiral polytopes of type $\{4,4,4\}$, with $1024m^2$ and $2048m^2$ automorphisms for every positive integer $m$, respectively. The automorphism groups of these polytopes are solvable groups, and when $m$ is a power of $2$, they provide examples with automorphism groups of order $2^n$ where $n$ can be any integer greater than $9$. (On the other hand, no chiral polytopes of type $[4,4,4]$ exist for $n \leq 9$.) In particular, our two families give a partial answer to a problem proposed by Schulte and Weiss in [Problems on polytopes, their gr…
This paper focuses on determining the volumes of permutation polytopes associated to cyclic groups, dihedral groups, groups of automorphisms of tree graphs, and Frobenius groups. We do this through the use of triangulations and the calculation of Ehrhart polynomials. We also present results on the theta body hierarchy of various permutation polytopes.
For each prime power $q\geq 5$, we construct a rank four chiral polytope that has a group $PSL(3,q)$ as automorphism group and Schläfli type $\{q-1,\frac{2(q-1)}{(3,q-1)},q-1\}$. We also construct rank five polytopes for some values of $q$ and we show that there is no chiral polytope of rank at least six having a group $PSL(3,q)$ or $PSU(3,q)$ as automorphism group.
Cayley extensions of maniplexes and polytopes
本文将Cayley地图概念推广至maniplexes与多面体，定义了Cayley扩张，证明文献中诸多自然扩张实为Cayley扩张，并考察其自同构群与对称型图。
A real representation of a finite group naturally determines a polytope, generalizing the well-known Birkhoff polytope. This paper determines the structure of the polytope corresponding to the natural permutation representation of a general Frobenius group.
For any given finite group, Schulte and Williams (2015) establish the existence of a convex polytope whose combinatorial automorphisms form a group isomorphic to the given group. We provide here a shorter proof for a stronger result: the convex polytope we build for the given finite group is binary, and even combinatorial in the sense of Naddef and Pulleyblank (1981); the diameter of its skeleton is at most 2; any combinatorial automorphism of the polytope is induced by some isometry of the space; any automorphism of the skeleton is a combinatorial automorphism.
We compute the set of facets of the polytope which is the convex hull of the Coxeter groups $\mathsf{F}_4$ or $\mathsf{H}_4$: For the group $\mathsf{F}_4$ we found $2$ orbits of facets which contradicts previous results published in \cite{birkhoff}. For the group $\mathsf{H}_4$ we found $1063$ orbits of facets which provides a counterexample to the conjecture of \cite{birkhoff}.
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.
We present a new algorithm to compute all the chiral polytopes that have a given group $G$ as full automorphism group. This algorithm uses a new set of generators that characterize the group, all of them except one being involutions. It permits to compute all chiral polytopes of groups that were previously unreachable by other known algorithms.
All polytopes are coset geometries: characterizing automorphism groups of k-orbit abstract polytopes
本文证明任意抽象n-多面体均可作为陪集几何构造，通过生成元、关系和交条件刻画了k-轨道多面体自同构群，并证明对所有k≠2及n≥3存在具有布尔自同构群的k-轨道n-多面体。
An abstract polytope of rank n is said to be chiral if its automorphism group has two orbits on the flags, such that adjacent flags belong to distinct orbits. Examples of chiral polytopes have been difficult to find. A "mixing" construction lets us combine polytopes to build new regular and chiral polytopes. By using the chirality group of a polytope, we are able to give simple criteria for when the mix of two polytopes is chiral.
Given two convex polytopes, the join, the cartesian product and the direct sum of them are well understood. In this paper we extend these three kinds of products to abstract polytopes and introduce a new product, called the topological product, which also arises in a natural way. We show that these products have unique prime factorization theorems. We use this to compute the automorphism group of a product in terms of the automorphism groups of the factors and show that (non trivial) products are almost never regular or two-orbit polytopes. We finish the paper by studying the monodromy group o…
It is already known that the automorphism group of a chiral polyhedron is never isomorphic to $PSL(2,q)$ or $PGL(2,q)$ for any prime power $q$. In this paper, we show that $PSL(2,q)$ and $PGL(2,q)$ are never automorphism groups of chiral polytopes of rank at least $5$. Moreover, we show that $PGL(2,q)$ is the automorphism group of at least one chiral polytope of rank $4$ for every $q\geq5$. Finally, we determine for which values of $q$ the group $PSL(2,q)$ is the automorphism group of a chiral polytope of rank $4$, except when $q=p^d\equiv3\pmod{4}$ where $d>1$ is not a prime power, in which c…
Up to isomorphism and duality, there are exactly two non-degenerate abstract regular polytopes of rank greater than $n-3$, one of rank $n-1$ and one of rank $n-2$, with automorphism groups that are transitive permutation groups of degree $n\geq 7$. In this paper we extend this classification of high rank regular polytopes to include the ranks $n-3$ and $n-4$. The result is, up to a isomorphism and duality, seven abstract regular polytopes of rank $n-3$ for each $n\geq 9$, and nine abstract regular polytopes of rank $n-4$ for each $n \geq 11$. Moreover we show that if a transitive permutation g…
Given a permutation group acting on coordinates of $\mathbb{R}^n$, we consider lattice-free polytopes that are the convex hull of an orbit of one integral vector. The vertices of such polytopes are called \emph{core points} and they play a key role in a recent approach to exploit symmetry in integer convex optimization problems. Here, naturally the question arises, for which groups the number of core points is finite up to translations by vectors fixed by the group. In this paper we consider transitive permutation groups and prove this type of finiteness for the $2$-homogeneous ones. We provid…
We investigate the combinatorics and geometry of permutation polytopes associated to cyclic permutation groups, i.e., the convex hulls of cyclic groups of permutation matrices. We give formulas for their dimension and vertex degree. In the situation that the generator of the group consists of at most two orbits, we can give a complete combinatorial description of the associated permutation polytope. In the case of three orbits the facet structure is already quite complex. For a large class of examples we show that there exist exponentially many facets.
The Ehrhart polynomial and the reciprocity theorems by Ehrhart \& Macdonald are extended to tensor valuations on lattice polytopes. A complete classification is established of tensor valuations of rank up to eight that are equivariant with respect to the special linear group over the integers and translation covariant. Every such valuation is a linear combination of the Ehrhart tensors which is shown to no longer hold true for rank nine.
Mirkovic and Vilonen discovered a canonical basis of algebraic cycles for the intersection homology of (the closures of the strata of) the loop Grassmannian. The moment map images of these varieties are a collection of polytopes, and they may be used to compute weight multiplicities and tensor product multiplicities for representations of a semisimple group. The polytopes are explicitly described for a few low rank groups.
We study modular and integral flow polynomials of graphs by means of subgroup arrangements and lattice polytopes. We introduce an Eulerian equivalence relation on orientations, flow arrangements, and flow polytopes; and we apply the theory of Ehrhart polynomials to obtain properties of modular and integral flow polynomials. The emphasis is on the geometrical treatment through subgroup arrangements and Ehrhart polynomials. Such viewpoint leads to a reciprocity law for the modular flow polynomial, which gives rise to an interpretation on the values of the modular flow polynomial at negative inte…
In the present paper, we give some sufficient conditions for $\operatorname{Cl}(\Bbbk[P])$ to be torsionfree, where $\operatorname{Cl}(\Bbbk[P])$ denote the divisor class group of the toric ring $\Bbbk[P]$ of an integral polytope $P$. We prove that $\operatorname{Cl}(\Bbbk[P])$ is torsionfree if $P$ is compressed, and $\operatorname{Cl}(\Bbbk[P])$ is torsionfree if $P$ is a $(0,1)$-polytope which has at most $\dim P+2$ facets. Moreover, we characterize the toric rings of $(0,1)$-polytopes in the case $\operatorname{Cl}(\Bbbk[P])\cong \mathbb{Z}$.
We study properties of the realizations of groups as the combinatorial automorphism group of a convex polytope. We show that for any non-abelian group $G$ with a central involution there is a centrally symmetric polytope with $G$ as its combinatorial automorphisms. We show that for each integer $n$, there are groups that cannot be realized as the combinatorial automorphisms of convex polytopes of dimension at most $n$. We also give an optimal lower bound for the dimension of the realization of a group as the group of isometries that preserves a convex polytope.
Rubik's Abstract Polytopes
本文将魔方及其构型群推广至任意抽象正多面体，完整刻画任意维魔方单纯形的相关群，并概述魔方超立方体的类似结论。
In this note we investigate the convex hull of those $n \times n$-permutation matrices that correspond to symmetries of a regular $n$-gon. We give the complete facet description. As an application, we show that this yields a Gorenstein polytope, and we determine the Ehrhart $h^*$-vector.
In this paper, we construct an infinite families of group $G$ of order $4p^m$ which can be an automorphism group of some regular polytope with type $\{p, 2p\}$, where $m \geq 3$ and $p$ is an odd prime. For $p=3$, our polytopes are the regular toroidal maps $\{3, 6\}$.
This is the second of two papers where we study polytopes arising from affine Coxeter arrangements. Our results include a formula for their volumes, and also compatible definitions of hypersimplices, descent numbers and major index for all Weyl groups. We give a q-analogue of Weyl's formula for the order of the Weyl group. For A_n, C_n and D_4, we give a Grobner basis which induces the triangulation of alcoved polytopes.
The Birkhoff polytope $B_n$ is the convex hull of all $n\times n$ permutation matrices in $\mathbb{R}^{n\times n}$. We compute the combinatorial symmetry group of the Birkhoff polytope. A representation polytope is the convex hull of some finite matrix group $G\leq \operatorname{GL}(d,\mathbb{R})$. We show that the group of permutation matrices is essentially the only finite matrix group which yields a representation polytope with the same face lattice as the Birkhoff polytope.
We augment the list of finite universal locally toroidal regular polytopes of type {3,3,4,3,3} due to P.McMullen and E.Schulte, adding as well as removing entries. This disproves a related long-standing conjecture. Our new universal polytope is related to a well-known Y-shaped presentation for the sporadic simple group $Fi_{22}$, and admits $S_4\times O_8^+(2){:}S_3$ as the automorphism group. We also discuss further extensions of its quotients in the context of Y-shaped presentations. As well, we note that two known examples of finite universal polytopes of type {3,3,4,3,3} are related to Y-s…
In this article, we research on the correspondences between the geometry of del Pezzo surfaces S_{r} and the geometry of Gosset polytopes (r-4)_{21}. We construct Gosset polytopes (r-4)_{21} in Pic S_{r}; Q whose vertices are lines, and we identify divisor classes in Pic S_{r} corresponding to (a-1)-simplexes, (r-1)-simplexes and (r-1)-crosspolytopes of the polytope (r-4)_{21}. Then we explain these classes correspond to skew a-lines, exceptional systems and rulings, respectively. As an application, we work on the monoidal transform for lines to study the local geometry of the polytope (r-4)_{…
Abstract Regular Polytopes of Finite Irreducible Coxeter Groups
本文研究了有限不可约Coxeter群的抽象正多面体，确定了其最大秩及不同秩的正多面体存在性。
When $G$ is a complex reductive algebraic group, MV polytopes are in bijection with the non-negative tropical points of the unipotent group of $G$. By fixing $w$ from the Weyl group, we can define MV polytopes whose highest vertex is labelled by $w$. We show that these polytopes are in bijection with the non-negative tropical points of the reduced double Bruhat cell labelled by $w^{-1}$. To do this, we define a collection of generalized minor functions $Δ_γ^\text{new}$ which tropicalize on the reduced Bruhat cell to the BZ data of an MV polytope of highest vertex $w$. We also describe the comb…
In this paper we show that the rank of every chiral polytope having a Suzuki group as automorphism group is $3$. This gives a positive answer to a conjecture of Isabel Hubard and Dimitri Leemans.
This is the first one of a series of papers on association of orientations, lattice polytopes, and abelian group arrangements to graphs. The purpose is to interpret the integral and modular tension polynomials of graphs at zero and negative integers. The whole exposition is put under the framework of subgroup arrangements and the application of Ehrhart polynomials. Such viewpoint leads to the following main results of the paper: (i) the reciprocity law for integral tension polynomials; (ii) the reciprocity law for modular tension polynomials; and (iii) a new interpretation for the value of the…
Lower affine MV polytopes of rank 2
本文证明秩2下仿射MV多面体与约化双Bruhat胞腔的非负热带点一一对应，推广了有限型结果，并因反射对称性同样适用于上仿射MV多面体。
We determine the diameter of the 1-skeleton and the combinatorial automorphism group of any Gelfand-Tsetlin polytope GT$_λ$ associated to an integer partition $λ.$
An abstract polytope is chiral if its automorphism group has two orbits on the flags, such that adjacent flags belong to distinct orbits. There are still few examples of chiral polytopes, and few constructions that can create chiral polytopes with specified properties. In this paper, we show how to build self-dual chiral polytopes using the mixing construction for polytopes.
Consider a lattice in a real finite dimensional vector space. Here, we are interested in the lattice polytopes, that is the convex hulls of finite subsets of the lattice. Consider the group $G$ of the affine real transformations which map the lattice onto itself. Replacing the group of euclidean motions by the group $G$ one can define the notion of regular lattice polytopes. More precisely, a lattice polytope is said to be regular if the subgroup of $G$ which preserves the polytope acts transitively on the set of its complete flags. Recently, Karpenkov obtained a classification of the regular …
A $k$-orbit maniplex is one that has $k$ orbits of flags under the action of its automorphism group. In this paper we extend the notion of symmetry type graphs of maps to that of maniplexes and polytopes and make use of them to study $k$-orbit maniplexes, as well as fully-transitive 3-maniplexes. In particular, we show that there are no fully-transtive $k$-orbit 3-mainplexes with $k > 1$ an odd number, we classify 3-orbit mainplexes and determine all face transitivities for 3- and 4-orbit maniplexes. Moreover, we give generators of the automorphism group of a polytope or a maniplex, given its …
String C-groups of 2-power order project onto a common string C-group
本文研究了2次幂阶的字符串C群的性质，发现了一个特殊的正则d-多面体C_d，其自同构群的阶为2^{2d-1}，并且是所有高阶正则d-多面体中具有最小旗数的唯一最小多面体，同时证明了该多面体的自同构群是所有有限2次幂阶字符串C群的商群，且所有具有2次幂阶自同构群的有限正则d-多面体都覆盖C_d。
Four infinite families of chiral $3$-polytopes of type $\{4, 8\}$ with solvable automorphism groups
本研究构造了四类具有可解自同构群的{4,8}型手性3-多胞体无限族，部分回应了Schulte与Weiss、Pellicer提出的相关问题。
Regular polytopes of rank $n/2$ for transitive groups of degree $n$
该研究针对次数为$n$的传递群对应的抽象正多胞体，探究秩为$n/2$（$n\neq14$）的情况，发现秩较最大值降低1时，正多胞体数量会大幅增加，补充了相关领域的研究结论。
Even with the introduction of supercharacter theories, the representation theory of many unipotent groups remains mysterious. This paper constructs a family of supercharacter theories for normal pattern groups in a way that exhibit many of the combinatorial properties of the set partition combinatorics of the full uni-triangular groups, including combinatorial indexing sets, dimensions, and computable character formulas. Associated with these supercharacter theories is also a family of polytopes whose integer lattice points give the theories geometric underpinnings.
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