Q-polynomial schemes, coherent configurations, and applications

University Of Wyoming, Laramie WY

Investigators

Abstract

Combinatorics is a broad area of mathematics that has found applications to many other fields such as computer science, statistics, physics, and chemistry. Association schemes and coherent configurations give a unified framework for several areas of combinatorics, such as coding theory, the statistical design of experiments, and finite geometry. This work has the potential to shed light on many problems in other areas of combinatorics, including those mentioned above as well as extremal graph theory. This project will further explore the rich connections between algebra and combinatorics, and help demarcate new directions, problems, and questions, thereby stimulating further interest in the area. Broader impacts include a sharpening of mathematical tools for applications in industry, training of highly qualified graduate students for academia and industry, and undergraduate research opportunities. The interaction between linear and abstract algebra and combinatorics has been a very fruitful area of study and continues to find applications beyond pure mathematics, in physics, computer science, and statistics. In this project, the PI and his team study association schemes and coherent configurations. The first part of the project is a study of association schemes with the so-called Q-polynomial property, a property formally dual to the notion of a distance-regular graph. While distance-regular graphs have been well studied, until the last decade little attention was paid to schemes with the Q-polynomial property that did not also arise from distance-regular graphs. Recent results suggest that these objects are of interest in and of themselves, and that the surprising structure of these schemes merits further exploration. The PI will continue the search for new examples of Q-polynomial schemes, with particular emphasis on those that arise from groups. The search will be complemented by work to characterize Q-polynomial schemes. The second part of the project concerns extending results of association schemes to the more general notion of coherent configurations, a natural generalization of association schemes. In particular, the PI will explore further applications of the recently discovered semidefinite bound in coherent configurations.