A Note on the Rank 5 Polytopes of M24

Orthogonal groups in characteristic 2 acting on polytopes of high rank

We show that for all integers $m\geq 2$, and all integers $k\geq 2$, the orthogonal groups $\Orth^{\pm}(2m,\Fk)$ act on abstract regular polytopes of rank $2m$, and the symplectic groups $\Sp(2m,\Fk)$ act on abstract regular polytopes of rank $2m+1$.

Projective linear groups as automorphism groups of chiral polytopes

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…

有限不可约Coxeter群的抽象正多面体

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。

对称多面体的2-群自同构群

Symmetric polytopes whose automorphism groups are 2-groups

研究了具有2-群自同构群的多面体，证明了其自同构群的继承性及扩展性质。

2次幂阶的字符串C群投射到共同的字符串C群

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。

偶特征下 $G_2(q)$ 的抽象正多胞形

Abstract regular polytopes of $G_2(q)$ in even characteristic

本文利用自然七维矩阵表示，显式构造了偶特征下 Chevalley 单群 $G_2(q)$ 的秩五字符串 C-群表示，并通过普查证明此类表示稀有且该构造具有例外性。

Polytopes associated to Dihedral Groups

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.

Reflection groups and polytopes over finite fields, II

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…

Groups of Ree type in characteristic 3 acting on polytopes

Every Ree group $R(q)$, with $q\neq 3$ an odd power of 3, is the automorphism group of an abstract regular polytope, and any such polytope is necessarily a regular polyhedron (a map on a surface). However, an almost simple group $G$ with $R(q) < G \leq \mathsf{Aut}(R(q))$ is not a C-group and therefore not the automorphism group of an abstract regular polytope of any rank.

Internal and external duality in abstract polytopes

We define an abstract regular polytope to be internally self-dual if its self-duality can be realized as one of its symmetries. This property has many interesting implications on the structure of the polytope, which we present here. Then, we construct many examples of internally self-dual polytopes. In particular, we show that there are internally self-dual regular polyhedra of each type $\{p, p\}$ for $p \geq 3$ and that there are both infinitely many internally self-dual and infinitely many externally self-dual polyhedra of type $\{p, p\}$ for $p$ even. We also show that there are internally…

双生成元单关系群的 JSJ 分解与多面体

JSJ decompositions and polytopes for two-generator one-relator groups

本文建立秩为2阿贝尔化的双曲双生成元单关系群的$Z_{max}$-JSJ分解与Friedl–Tillmann多面体的直接联系，并导出二次时间算法等结构性质。