On algebraic fusions of association schemes

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representations of a Schur cover of the corresponding group of algebraic automorphisms.Comment: This paper has been withdrawn by one of the authors, since it requires more wor

Four-class Skew-symmetric Association Schemes

An association scheme is called skew-symmetric if it has no symmetric adjacency relations other than the diagonal one. In this paper, we study 4-class skew-symmetric association schemes. In J. Ma [On the nonexistence of skew-symmetric amorphous association schemes, submitted for publication], we discovered that their character tables fall into three types. We now determine their intersection matrices. We then determine the character tables and intersection numbers for 4-class skew-symmetric pseudocyclic association schemes, the only known examples of which are cyclotomic schemes. As a result, we answer a question raised by S. Y. Song [Commutative association schemes whose symmetrizations have two classes, J. Algebraic Combin. 5(1) 47-55, 1996]. We characterize and classify 4-class imprimitive skew-symmetric association schemes. We also prove that no 2-class Johnson scheme can admit a 4-class skew-symmetric fission scheme. Based on three types of character tables above, a short list of feasible parameters is generated.Comment: 12 page

Modelling the Developing Mind: From Structure to Change

This paper presents a theory of cognitive change. The theory assumes that the fundamental causes of cognitive change reside in the architecture of mind. Thus, the architecture of mind as specified by the theory is described first. It is assumed that the mind is a three-level universe involving (1) a processing system that constrains processing potentials, (2) a set of specialized capacity systems that guide understanding of different reality and knowledge domains, and (3) a hypecognitive system that monitors and controls the functioning of all other systems. The paper then specifies the types of change that may occur in cognitive development (changes within the levels of mind, changes in the relations between structures across levels, changes in the efficiency of a structure) and a series of general (e.g., metarepresentation) and more specific mechanisms (e.g., bridging, interweaving, and fusion) that bring the changes about. It is argued that different types of change require different mechanisms. Finally, a general model of the nature of cognitive development is offered. The relations between the theory proposed in the paper and other theories and research in cognitive development and cognitive neuroscience is discussed throughout the paper

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

Uniformity in association schemes and coherent configurations: cometric Q-antipodal schemes and linked systems

Inspired by some intriguing examples, we study uniform association schemes and uniform coherent configurations, including cometric Q-antipodal association schemes. After a review of imprimitivity, we show that an imprimitive association scheme is uniform if and only if it is dismantlable, and we cast these schemes in the broader context of certain --- uniform --- coherent configurations. We also give a third characterization of uniform schemes in terms of the Krein parameters, and derive information on the primitive idempotents of such a scheme. In the second half of the paper, we apply these results to cometric association schemes. We show that each such scheme is uniform if and only if it is Q-antipodal, and derive results on the parameters of the subschemes and dismantled schemes of cometric Q-antipodal schemes. We revisit the correspondence between uniform indecomposable three-class schemes and linked systems of symmetric designs, and show that these are cometric Q-antipodal. We obtain a characterization of cometric Q-antipodal four-class schemes in terms of only a few parameters, and show that any strongly regular graph with a ("non-exceptional") strongly regular decomposition gives rise to such a scheme. Hemisystems in generalized quadrangles provide interesting examples of such decompositions. We finish with a short discussion of five-class schemes as well as a list of all feasible parameter sets for cometric Q-antipodal four-class schemes with at most six fibres and fibre size at most 2000, and describe the known examples. Most of these examples are related to groups, codes, and geometries.Comment: 42 pages, 1 figure, 1 table. Published version, minor revisions, April 201

Decomposition algebras and axial algebras

We introduce decomposition algebras as a natural generalization of axial algebras, Majorana algebras and the Griess algebra. They remedy three limitations of axial algebras: (1) They separate fusion laws from specific values in a field, thereby allowing repetition of eigenvalues; (2) They allow for decompositions that do not arise from multiplication by idempotents; (3) They admit a natural notion of homomorphisms, making them into a nice category. We exploit these facts to strengthen the connection between axial algebras and groups. In particular, we provide a definition of a universal Miyamoto group which makes this connection functorial under some mild assumptions. We illustrate our theory by explaining how representation theory and association schemes can help to build a decomposition algebra for a given (permutation) group. This construction leads to a large number of examples. We also take the opportunity to fix some terminology in this rapidly expanding subject.Comment: 23 page