Commutative association schemes

Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the "commutative case," has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work.Comment: 36 page

Noncommutative reality-based algebras of rank 6

We show that noncommutative standard reality-based algebras (RBAs) of dimension 6 are determined up to exact isomorphism by their character tables. We show that the possible character tables of these RBAs are determined by seven real numbers, the first four of which are positive and the remaining three real numbers can be arbitrarily chosen up to a single exception. We show how to obtain a concrete matrix realization of the elements of the RBA-basis from the character table. Using a computer implementation, we give a list of all noncommutative integral table algebras of rank 6 with orders up to 150. Four in the list are primitive, but we show three of them cannot be realized as adjacency algebras of association schemes. In the last section of the paper, we apply our methods to give a precise description of the noncommutative integral table algebras of rank 6 for which the multiplicity of both linear characters is 1

Noncommutative Reality-based Algebras of Rank 6

We show that noncommutative standard reality-based algebras (RBAs) of dimension 6 are determined up to exact isomorphism by their character tables. We show that the possible character tables of these RBAs are determined by seven real numbers, the first four of which are positive and the remaining three real numbers can be arbitrarily chosen up to a single exception. We show how to obtain a concrete matrix realization of the elements of the RBA-basis from the character table. Using a computer implementation, we give a list of all noncommutative integral table algebras of rank 6 with orders up to 150. Four in the list are primitive, but we show three of them cannot be realized as adjacency algebras of association schemes. In the last section of the paper, we apply our methods to give a precise description of the noncommutative integral table algebras of rank 6 for which the multiplicity of both linear characters is 1

Permutation group approach to association schemes

AbstractWe survey the modern theory of schemes (coherent configurations). The main attention is paid to the schurity problem and the separability problem. Several applications of schemes to constructing polynomial-time algorithms, in particular, graph isomorphism tests, are discussed

The recognition problem for table algebras and reality-based algebras

Given a finite-dimensional noncommutative semisimple algebra $A$ with involution, we show that $A$ always has an RBA-basis. We look for an RBA-basis that has integral or rational structure constants, and ask if the RBA admits a positive degree map. For RBAs that have a positive degree map, we try to find an RBA-basis with nonnegative structure constants to determine if there is a generalized table algebra structure. We settle these questions for the algebras $\mathbb{C} \oplus M_n(\mathbb{C})$, $n \ge 2$.Comment: 16 page

有限射影幾何の組合せ構造の代数的性質について

Tohoku University田中太初課

Partial geometric designs and difference families

We examine the designs produced by different types of difference families. Difference families have long been known to produce designs with well behaved automorphism groups. These designs provide the elegant solutions desired for applications. In this work, we explore the following question: Does every (named) design have a difference family analogue? We answer this question in the affirmative for partial geometric designs