Related papers: Non-very generic arrangements in low dimension
Hyperplane arrangements form the geometric counterpart of combinatorial objects such as matroids. The shape of the sequence of Betti numbers of the complement of a hyperplane arrangement is of particular interest in combinatorics, where…
Bayesian mixture models are widely used for clustering of high-dimensional data with appropriate uncertainty quantification. However, as the dimension of the observations increases, posterior inference often tends to favor too many or too…
The set of chambers of a real hyperplane arrangement may be ordered by separation from some fixed chamber. When this poset is a lattice, Bjorner, Edelman, and Ziegler proved that the chambers are in natural bijection with the biconvex sets…
For a rank two root system and a pair of nonnegative integers, using only elementary combinatorics we construct two posets. The constructions are uniform across the root systems A1+A1, A2, C2, and G2. Examples appear in Figures 3.2 and 3.3.…
A configuration of lattice vectors is supernormal if it contains a Hilbert basis for every cone spanned by a subset. We study such configurations from various perspectives, including triangulations, integer programming and Groebner bases.…
Linde, Moore, and Nordahl introduced a generalisation of the honeycomb dimer model to higher dimensions. The purpose of this article is to describe a number of structural properties of this generalised model. First, it is shown that the…
Cubic bipartite graphs with girth at least 6 correspond to symmetric combinatorial (v_3) configurations. In 1887 V. Martinetti described a simple reduction method which enables one to reduce each combinatorial (v_3) configuration to one…
Algebra bundles, in the strict sense, appear in many areas of geometry and physics. However, the structure of an algebra is flexible enough to vary non-trivially over a connected base, giving rise to a structure of a weak algebra bundle. We…
The rings of SL(V) invariants of configurations of vectors and linear forms in a finite-dimensional complex vector space V were explicitly described by Hermann Weyl in the 1930s. We show that when V is 3-dimensional, each of these rings…
An $(n_k)$-configuration is a set of $n$ points and $n$ lines in the projective plane such that their point-line incidence graph is $k$-regular. The configuration is geometric, topological, or combinatorial depending on whether lines are…
In [9], [15] it has been introduced a technique, based on nonstandard analysis, to study some problems in combinatorial number theory. In this paper we present three applications of this technique: the first one is a new proof of a known…
In 1970 A. Baker and W. Schmidt introduced regular systems of numbers and vectors, showing that the set of real algebraic numbers forms a regular system on any fixed interval. This fact was used to prove several important results in the…
In this article, we study permanental varieties, i.e. varieties defined by the vanishing of permanents of fixed size of a generic matrix. Permanents and their varieties play an important, and sometimes poorly understood, role in…
In 1941, Brauer-Nesbitt established a characterization of a block with trivial defect group as a block $B$ with $k(B) = 1$ where $k(B)$ is the number of irreducible ordinary characters of $B$. In 1982, Brandt established a characterization…
Very Special Relativity (VSR) framework, proposed by Cohen and Glashow [1], demonstrated that a proper subgroup of the Poincar\'e group, (in particular ISIM(2)), is sufficient to describe the spacetime symmetries of the so far observed…
In this paper, we study $k$-parabolic arrangements, a generalization of $k$-equal arrangements for finite real reflection groups. When $k=2$, these arrangements correspond to the well-studied Coxeter arrangements. Brieskorn (1971) showed…
Let $K$ be a nonarchimedean local field of characteristic zero with valuation ring $R$, for instance, $K=\mathbb{Q}_p$ and $R=\mathbb{Z}_p$. We prove a general integral geometric formula for $K$-analytic groups and homogeneous $K$-analytic…
We study supersolvable line arrangements in ${\mathbb P}^2$ over the reals and over the complex numbers, as the first step toward a combinatorial classification. Our main results show that a nontrivial (i.e., not a pencil or near pencil)…
$\renewcommand{\Re}{\mathbb{R}}$ We re-examine parameters for the two main space decomposition techniques---bottom-vertex triangulation, and vertical decomposition, including their explicit dependence on the dimension $d$, and discover…
We introduce lattice gauge theories which describe three-dimensional, gapped quantum phases exhibiting the phenomenology of both conventional three-dimensional topological orders and fracton orders, starting from a finite group $G$, a…