Related papers: Petrie-Coxeter Maps Revisited

We construct a new monoid structure for Artin groups associated with finite Coxeter systems. This monoid shares with the classical positive braid monoid a crucial algebraic property: it is a Garside monoid. The analogy with the classical…

A biconvex polytope is a classical and tropical convex hull of finitely many points. Given a biconvex polytope, for each vertex of it we construct a directed bigraph and a gammoid so that the collection of base polytopes of those gammoids…

In this article we study the cohomology of deep level Deligne--Lusztig varieties of Coxeter type, attached to a reductive group over a local non-archimedean field, which splits over an unramified extension. This allows to construct some new…

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…

For a path-connected metric space $(X,d)$, the $n$-th homotopy group $\pi_n(X)$ inherits a natural pseudometric from the $n$-th iterated loop space with the uniform metric. This pseudometric gives $\pi_n(X)$ the structure of a topological…

We present a relation between volumes of certain lower dimensional simplices associated to a full-dimensional primal and polar dual polytope in R^k. We then discuss an application of this relation to a geometric construction of a Colin de…

In this note we first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order)…

We propose a classification of polyhedra (planar, $3$-connected graphs) according to their type i.e., their set of quantities of common neighbours for each pair of distinct vertices. For every (finite) set of non-negative integers, we…

We investigate small covers and quasitoric over the duals of neighborly simplicial polytopes with small number of vertices in dimensions $4$, $5$, $6$ and $7$. In the most of the considered cases we obtain the complete classification of…

A Coxeter polytope is a convex polytope in a real projective space equipped with linear reflections in its facets, such that the orbits of the polytope under the action of the group generated by the linear reflections tessellate a convex…

Quaternionic representations of Coxeter (reflection) groups of ranks 3 and 4, as well as those of E_8, have been used extensively in the literature. The present paper analyses such Coxeter groups in the Clifford Geometric Algebra framework,…

We compute Aut(W) for any even Coxeter group whose Coxeter diagram is connected, contains no edges labeled 2, and cannot be separated into more than 2 connected components by removing a single vertex. The description is given explicitly in…

Marginal polytopes are important geometric objects that arise in statistics as the polytopes underlying hierarchical log-linear models. These polytopes can be used to answer geometric questions about these models, such as determining the…

We present a higher index theorem for a certain class of etale one-dimensional complex-analytic groupoids. The novelty is the use of the local anomaly formula established in a previous paper, which represents the bivariant Chern character…

This is a chapter in an upcoming Tamari Festscrift. Permutahedra are a class of convex polytopes arising naturally from the study of finite reflection groups, while generalized associahedra are a class of polytopes indexed by finite…

In the 19th International Symposium on Advances in Robot Kinematics the author introduced a novel class of continuous flexible discrete surfaces and mentioned that these so-called P-hedra (or P-nets) allow direct access to their spatial…

In this work, following the Discrete de Rham (DDR) paradigm, we develop an arbitrary-order discrete divdiv complex on general polyhedral meshes. The construction rests 1) on discrete spaces that are spanned by vectors of polynomials whose…

For typical cocycles over subshifts of finite type, we show that for any given orbit segment, we can construct a periodic orbit such that it shadows the given orbit segment and that the product of the cocycle along its orbit is a proximal…

A product of cochains in a polyhedral complex is constructed. The multiplication algorithm depends on the choice of a parameter. The parameter is a linear functional on the ambient space. Cocycles form a subring of the ring of cochains,…

Knowing the symmetries of a polyhedron can be very useful for the analysis of its structure as well as for practical polyhedral computations. In this note, we study symmetry groups preserving the linear, projective and combinatorial…