Related papers: Petrie-Coxeter Maps Revisited
Starting from a finite simple graph $G$, for each eigenvalue $\theta$ of its adjacency matrix one can construct a convex polytope $P_G(\theta)$, the so called $\theta$-eigenpolytop of $G$. For some polytopes this technique can be used to…
We study quartic surfaces that admit a group of projective automorphisms isomorphic to icosahedron group.
The topic of the paper are developments of $n$-dimensional Coxeter polyhedra. We show that the surface of such polyhedron admits a canonical cutting such that each piece can be covered by a Coxeter $(n-1)$-dimensional domain.
Howe and Tan (1993) investigated a degenerate principal series representation of indefinite orthogonal groups $\mathrm{O}(V)$ and explicitly described its composition series. They showed that there exists a unique unitarizable irreducible…
The aim of this paper is to study alcoved polytopes, which are polytopes arising from affine Coxeter arrangements. This class of convex polytopes includes many classical polytopes, for example, the hypersimplices. We compare two…
We construct exotic copies of $\mathbb{R}^4$ with nontrivial compactly supported mapping class groups of arbitrarily large rank. This follows from a modification of the construction of the diffeomorphism corks of arXiv:2407.04696 that makes…
Self-polar polytopes are convex polytopes that are equal to an orthogonal transformation of their polar sets. These polytopes were first studied by Lov\'{a}sz as a means of establishing the chromatic number of distance graphs on spheres,…
Using the Guirardel-Levitt outer space of a free product, we prove that the outer automorphism group of the outer automorphism group of the universal Coxeter group of rank $n \geq 5$ is trivial, and that it is a cyclic group of order 2 if…
The tree complex is a simplicial complex defined in recent work of Belk, Lanier, Margalit, and Winarski with natural applications to mapping class groups and complex dynamics. In this article, we connect this setting with the study of…
We define and study a new family of polytopes which are formed as convex hulls of partial alternating sign matrices. We determine the inequality descriptions, number of facets, and face lattices of these polytopes. We also study partial…
Discovery of clusters with high symmetrical geometry, such as C60 fullerene, always attract lots of interest because of their diverse nature. However, most of such interesting cluster were sporadically discovered, is there any systematic…
In this paper, we define (reduced) homeology groups and (reduced) cohomeology groups on finite simpicial complexes and prove that these groups are PL homeomorphsm invariants of polyhedra, while they are not homotopy invariants. So these…
We present the construction of Inoue surfaces of type $S^{(+)}/S^{(-)}$ in terms of data arising from real quadratic number fields. We then describe the automorphism group of such surfaces in terms of this data.
The associahedron is a convex polytope whose face poset is based on nonintersecting diagonals of a convex polygon. In this paper, given an arbitrary simple polygon P, we construct a polytopal complex analogous to the associahedron based on…
We consider the random hypergraph on a finite vertex set by choosing each set of vertices as an hyperedge independently at random. We express the probability distributions of the (lower-)associated simplicial complex and the…
By applying Borcherds' theory of automorphic forms on bounded symmetric domains of type IV, we give a 5-dimensional linear system of automorphic forms of weight 6 on Igusa quartic 3-fold which induces an S_6-equivariant rational map of…
We describe the cone of deformations of a Coxeter permutahedron, or equivalently, the nef cone of the toric variety associated to a Coxeter complex. This family of polytopes contains polyhedral models for the Coxeter-theoretic analogs of…
Let X be a space of constant curvature and P be a convex polyhedron in X. A Coxeter decomposition of the polyhedron P is a decomposition of P into finitely many Coxeter polyhedra, such that any two polyhedra having a common facet are…
In this paper, we show how regular convex 4-polytopes - the analogues of the Platonic solids in four dimensions - can be constructed from three-dimensional considerations concerning the Platonic solids alone. Via the Cartan-Dieudonne…
The paper describes a construction of abstract polytopes from Cayley graphs of symmetric groups. Given any connected graph G with p vertices and q edges, we associate with G a Cayley graph of the symmetric group S_p and then construct a…