球面$t$ -设计的概念是由Delsarte-Goethals-Seidel（1977）提出的，它是单位球面的有限子集。Euclidean $t$ -设计的概念，它是Neumaier-Seidel（1988）的球形设计的两步推广，在这个意义上它是欧几里得空间的有限加权子集。我们首先回顾这两个概念，以及紧密$t$设计的概念，即，其基数达到自然下界的那个。我们感兴趣的是$t$ -设计（球形或欧几里德），这是紧或接近紧。正如Delsarte-Goethals-Seidel（1977）所熟知的，在研究球形$t$ -设计，特别是紧设计或接近紧设计时，结合方案起着重要的作用。本文的主要目的是表明，在研究欧几里德$t$ -设计，特别是那些无论是紧或接近紧，相干配置发挥重要作用。这里，相干构形是由D. G. Higman提出的结合方案是通过公理化一般的、不一定是传递的置换群的性质而得到的，其方法与公理化传递的置换群的性质而得到的结合方案相同。本文的主要目的是证明满足一定条件的欧氏$t$ -设计给出相干组态的结构。在此基础上，研究了在一定的附加条件下两个同心球面上欧氏4-设计的分类问题。
英文摘要：
The concept of spherical $t$-design, which is a finite subset of the unit sphere, was introduced by Delsarte-Goethals-Seidel (1977). The concept of Euclidean $t$-design, which is a two step generalization of spherical design in the sense that it is a finite weighted subset of Euclidean space, by Neumaier-Seidel (1988). We first review these two concepts, as well as the concept of tight $t$-design, i.e., the one whose cardinality reaches the natural lower bound. We are interested in $t$-designs (spherical or Euclidean) which are either tight or close to tight. As is well known by Delsarte-Goethals-Seidel (1977), in the study of spherical $t$-designs and in particular of those which are either tight or close to tight, association schemes play important roles. The main purpose of this paper is to show that in the study of Euclidean $t$-designs and in particular of those which are either tight or close to tight, coherent configurations play important roles. Here, coherent configuration is a purely combinatorial concept defined by D. G. Higman, and is obtained by axiomatizing the properties of general, not necessarily transitive, permutation groups, in the same way as association scheme was obtained by axiomatizing the properties of transitive permutation groups. The main purpose of this paper is to prove that Euclidean $t$-designs satisfying certain conditions give the structure of coherent configurations. Moreover we study the classification problems of Euclidean 4-designs on two concentric spheres with certain additional conditions.
Finding the maximum cardinality of a $2$-distance set in Euclidean space is a classical problem in geometry. Lisoněk in 1997 constructed a maximum $2$-distance set in $\mathbb R^8$ with $45$ points. That $2$-distance set constructed by Lisoněk has a distinguished structure of a coherent configuration of type $(2,2;3)$ and is embedded in two concentric spheres in $\mathbb R^8$. In this paper we study whether there exists any other similar embedding of a coherent configuration of type $(2,2;3)$ as a $2$-distance set in $\mathbb R^n$, without assuming any restriction on the size of the set. We pr…
In this paper, we introduce Euclidean Gallai-Ramsey theory, by combining Euclidean Ramsey theory and Gallai-Ramsey theory on graphs. More precisely, we consider the following problem: For an integer $r$ and configurations $K$ and $K'$, does there exist an integer $n_0$ such that for any $r$-coloring of the points of $n$-dimensional Euclidean space with $n \geq n_0$, there is a monochromatic configuration congruent to $K$ or a rainbow configuration congruent to $K'$? In particular, we give a bound on $n_0$ for some configurations $K$ and $K'$, such as triangles and rectangles. Those are extensi…
Coherent configurations are a generalization of association schemes. In this paper, we introduce the concept of $Q$-polynomial coherent configurations and study the relationship among intersection numbers, Krein numbers, and eigenmatrices. The examples of $Q$-polynomial coherent configurations are provided from Delsarte designs in $Q$-polynomial schemes and spherical designs.
For fiber-commutative coherent configurations, we show that Krein parameters can be defined essentially uniquely. As a consequence, the general Krein condition reduces to positive semidefiniteness of finitely many matrices determined by the parameters of a coherent configuration. We mention its implications in the coherent configuration defined by a generalized quadrangle. We also simplify the absolute bound using the matrices of Krein parameters.
In 1988, D. G. Higman gave properties of parameters of strongly regular designs without proof. It included an error. We will give proofs and correct the error.
The minimal degree of a permutation group $G$ is the minimum number of points not fixed by non-identity elements of $G$. Lower bounds on the minimal degree have strong structural consequences on $G$. In 2014 Babai proved that the automorphism group of a strongly regular graph with $n$ vertices has minimal degree $\geq c n$, with known exceptions. Strongly regular graphs correspond to primitive coherent configurations of rank 3. We extend Babai's result to primitive coherent configurations of rank 4. We also show that the result extends to non-geometric distance-regular graphs of bounded diamet…
On circulant ternary coherent configurations of prime degree
Hainan University（海南大学）；Guangxi Technological College of Machinery and Electricity（广西机电职业技术学院）；Hebei Normal University（河北师范大学）；Steklov Institute of Mathematics at St. Petersburg（圣彼得堡斯捷克洛夫数学研究所）
Faculty of Mathematics, Kyushu University（九州大学数学系）；Department of Mathematics, National Defense Academy of Japan（日本防卫大学校数学系）；School of Mathematics, University of Shanghai for Science and Technology（上海理工大学数学学院）
A Cartesian decomposition of a coherent configuration $\cal X$ 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 $\cal X$ comes from a certain Cartesian decomposition. It is proved that if the coherent configuration $\cal X$ is thick, then there is a unique maximal Cartesian decomposition of $\cal X$, i.e., there is exactly one internal tensor decomposition of $\cal X$ into indecomposable components. In particular, this implies an analog of the Krull--Schmidt theorem for the thick coherent co…
We study a class of edge-coloured graphs, including the chamber systems of buildings and other geometries such as affine planes, from which we build coherent configurations (also known as non-commutative association schemes). The condition we require is that the graph be endowed with a certain distance function, taking its values in the adjacency algebra (itself generated by the adjacency operators). When all the edges are of the same colour, the condition is equivalent to the graph being distance-regular, so our result is a generalization of the classical fact that distance-regular graphs giv…
The minimal degree of a permutation group $G$ is the minimum number of points not fixed by non-identity elements of $G$. Lower bounds on the minimal degree have strong structural consequences on $G$. Babai conjectured that if a primitive coherent configuration with $n$ vertices is not a Cameron scheme, then its automorphism group has minimal degree $\geq cn$ for some constant $c>0$. In 2014, Babai proved the desired lower bound on the minimal degree of the automorphism groups of strongly regular graphs, thus confirming the conjecture for primitive coherent configurations of rank 3. In this pap…
Let $\mathbf{p}$ be a configuration of $n$ points in $\mathbb{R}^d$ for some $n$ and some $d \ge 2$. Each pair of points defines an edge, which has a Euclidean length in the configuration. A path is an ordered sequence of the points, and a loop is a path that has the same endpoints. A path or loop, as a sequence of edges, also has a Euclidean length. In this paper, we study the question of when $\mathbf{p}$ will be uniquely determined (up to an unknowable Euclidean transform) from a given set of path or loop lengths. In particular, we consider the setting where the lengths are given simply as …
We present a Euclidean quantum gravity model in which random graphs dynamically self-assemble into discrete manifold structures. Concretely, we consider a statistical model driven by a discretisation of the Euclidean Einstein-Hilbert action; contrary to previous approaches based on simplicial complexes and Regge calculus our discretisation is based on the Ollivier curvature, a coarse analogue of the manifold Ricci curvature defined for generic graphs. The Ollivier curvature is generally difficult to evaluate due to its definition in terms of optimal transport theory, but we present a new exact…
