Kaleidoscopical configurations

Let G be a group, and let X be a G-space with the action G × X → X, (g, x) ?gx. A subset A of X is called a kaleidoscopical configuration if there is a coloring χ : X → κ (i.e., a mapping of X onto a cardinal κ) such that the restriction χ|gA is a bijection for each g ∈ G. We survey some recent results on kaleidoscopical configurations in metric spaces considered as G-spaces with respect to the groups of their isometries and in groups considered as left regular G-spaces.     

Naive configurations

We describe a new way to construct finite geometric objects. For every \(k\) we obtain a symmetric configuration \(\mathcal{E }(k-1)\) with \(k\) points on a line. In particular, we have a constructive existence proof for such configurations. The method is very simple and purely geometric. It also produces interesting periodic matrices.     

Combinatorial configurations

Book Reviews

Combinatorial configurationsVladimir D. Tonchev: (Translated from Bulgarian to English by Robert A. Melter.) Pitman Monographs and Surveys in Pure and Applied Mathematics #40, Longman Scientific and Technical, Harlow, Essex, U.K., 1988, 189 pp.     

Canonical configurations

Given a finite configuration E in Rⁿ and an arbitrary coloring of Rⁿ (possibly with an infinite number of colors) it is shown that there exists E′ homothetic to E such that E′ is colored in a “special” way. The proof is based on the recent theorem of W. Deuber, R. L. Graham, H. J. Prömel, and B. Voigt, J. Combin. Theory Ser. A.34 (1983), 331–339.     

On convergence of configurations

The transition rule F of a cellular automaton may sometimes be regarded as a “rule of growth” of a crystal from a “seed” ω. A study is made of the iterates ω, Fω, F²ω,…. For certain growth rules F it is proved that when ω is “sufficiently large” the sequence F^pω “converges” to a rational polytope W. The limiting shape W depends only on F.     

Sur quelques configurations

Sans résumé     

Cartan coherent configurations

The Cartan scheme \(\mathcal{X}\) of a finite group G with a (B, N)-pair is defined to be the coherent configuration associated with the action of G on the right cosets of the Cartan subgroup \(B\cap N\) by right multiplication. It is proved that if G is a simple group of Lie type, then asymptotically the coherent configuration \(\mathcal{X}\) is 2-separable, i.e., the array of 2-dimensional intersection numbers determines \(\mathcal{X}\) up to isomorphism. It is also proved that in this case, the base number of \(\mathcal{X}\) equals 2. This enables us to construct a polynomial-time algorithm for recognizing Cartan schemes when the rank of G and the order of the underlying field are sufficiently large. One of the key points in the proof is a new sufficient condition for an arbitrary homogeneous coherent configuration to be 2-separable.     

Regularity of line configurations

We show that in arithmetically-Gorenstein line arrangements with only planar singularities, each line intersects the same number of other lines. This number has an algebraic interpretation: it is the Castelnuovo–Mumford regularity of the coordinate ring of the arrangement.We also prove that every $(d?1)$-dimensional simplicial complex whose 0-th and 1-st homologies are trivial is the nerve complex of a suitable d-dimensional standard graded algebra of depth ≥3. This provides the converse of a recent result by Katzman, Lyubeznik and Zhang.     

Minimum blocking configurations

Some results on configurations realizing minimum blocking sets of a finite projective plane are obtained by introducing a suitable attraction property.Dedicated to Professor M. Scafati Tallini on the occasion of her 65th birthday     

Large tactical configurations

It is shown that if r = 2, then for all m, n, h ≥ 3 and all “large enough” R, W such that mR = nW, there is a tactical configuration of rank 2 girth g = 2h, and degrees m and n on sets of cardinalities R and W. We also show that if r ≥ 3 then for all h and all compatible degree sets N = {n(i, j)≥3} and all large enough numbers R(1), R(2),…, R(r) satisfying R(i)n(i, j) = R(j)n(j, i) there is a tactical configuration of rank r, girth h, and degrees N on set with cardinalities R(1), R(2),…, R(r).     

The semigroup of combinatorial configurations

We elaborate on the existence and construction of the so-called combinatorial configurations. The main result is that for fixed degrees the existence of such configurations is given by a numerical semigroup. The proof is constructive giving a method to obtain combinatorial configurations with parameters large enough.     

Interactive generalizing of geometric configurations

The heuristic method of generalizing in the sense of theorem finding through conceptual generalization is a productive scheme of cognition which, in the field of synthetic elementary geometry, can be acquired and practised by the student in a direct way. With the computer as an interactive construction tool motivating and efficient possibilities arise to realize these methods.     

Tensor products of coherent configurations

A Cartesian decomposition of a coherent configuration ✗ is defined as a special set of its parabolics that form a Cartesian decomposition of the underlying set. It turns out that every tensor decomposition of ✗ comes from a certain Cartesian decomposition. It is proved that if the coherent configuration ✗ is thick, then there is a unique maximal Cartesian decomposition of ✗; i.e., there is exactly one internal tensor decomposition of ✗ into indecomposable components. In particular, this implies an analog of the Krull–Schmidt theorem for the thick coherent configurations. A polynomial-time algorithm for finding the maximal Cartesian decomposition of a thick coherent configuration is constructed.