Related papers: A survey on association schemes on triples
For any finite group $G$, and any positive integer $n$, we construct an association scheme which admits the diagonal group $D_n(G)$ as a group of automorphisms. The rank of the association scheme is the number of partitions of $n$ into at…
We give a complete description of the irreducible representations of algebraic fusions of association schemes, in terms of the irreducible representations of a Schur cover of the corresponding group of algebraic automorphisms.
Inspired by the ideas and techniques used in the study of cluster algebras we construct a new class of algebras, called bistellar cluster algebras, from closed oriented triangulated even-dimensional manifolds by performing…
B. Xu characterized rank $4$ self-dual association schemes inducing three partial geometric designs by their character tables. We construct such association schemes as Schur rings over abelian $2$-groups.
Freudenthal triple systems come in two flavors, degenerate and nondegenerate. The best criterion for distinguishing between the two which is available in the literature is by descent. We provide an identity which is satisfied only by…
An imprimitive symmetric indecomposable association scheme of rank 5 is said to be Higmanian. A divisible design graph is a graph whose adjacency matrix is an incidence matrix of a symmetric divisible design. We establish conditions which…
In this paper, we consider associative algebras equipped with derivations. A pair consisting of an associative algebra and a distinguished derivation is called an AssDer pair. We study central extensions and formal one-parameter…
In this paper, we construct directed strongly regular graphs and divisible design graphs with new parameters merging some basic relations of so-called Tatra associations schemes. We also study the above association schemes, their fusions…
Let $\mathbb{F}$ be an arbitrary field. In this paper, we continue studying the Terwilliger algebras of association schemes over $\mathbb{F}$ that were called the Terwilliger $\mathbb{F}$-algebras of association schemes in [8]. We determine…
Delsarte theory, more specifically the study of codes and designs in association schemes, has proved invaluable in studying an increasing assortment of association schemes in recent years. Tools motivated by the study of error-correcting…
Recent classification of $\frac{3}{2}$-transitive permutation groups leaves us with three infinite families of groups which are neither $2$-transitive, nor Frobenius, nor one-dimensional affine. The groups of the first two families…
Accidental symmetries in effective field theories can be established by computing and comparing Hilbert series. This invites us to study them with the tools of invariant theory. Applying this technology, we spotlight three classes of…
In this paper, the association scheme defined on the flags of a finite generalized quadrangle is considered. All possible fusions of this scheme are listed, and a full description for those of classes 2 and 3 is given. Furthermore, it is…
The Terwilliger algebras of association schemes over an arbitrary field $\mathbb{F}$ were called the Terwilliger $\mathbb{F}$-algebras of association schemes in [8]. In this paper, we study the Terwilliger $\mathbb{F}$-algebras of factorial…
We present a survey of recent results, scattered in a series of papers that appeared during past five years, whose common denominator is the use of cubic relations in various algebraic structures. Cubic (or ternary) relations can represent…
In this paper we determine all fusions of the association scheme $\mathcal{A} \otimes \mathcal{A}$, where $\mathcal{A}$ is the symmetric rank $3$ association scheme corresponding to a strongly regular graph. This includes both guaranteed…
In this paper, we study association schemes on the anisotropic points of classical polar spaces. Our main result concerns non-degenerate elliptic and hyperbolic quadrics in PG$(n,q)$ with $q$ odd. We define relations on the anisotropic…
The Terwilliger algebras of asymmetric association schemes of rank $3$, whose nonidentity relations correspond to doubly regular tournaments, are shown to have thin irreducible modules, and to always be of dimension $4k+9$ for some positive…
Association schemes are combinatorial objects that allow us solve problems in several branches of mathematics. They have been used in the study of permutation groups and graphs and also in the design of experiments, coding theory, partition…
A new class of random spatial tessellations is introduced -- the so-called column tessellations of three-dimensional space. The construction is based on a stationary planar tessellation. Each cell of the spatial tessellation is a prism…