Algebraic combinatorics and representation theory

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The proposed area of research is algebraic combinatorics, a highly active field that mixes algebraic and combinatorial techniques to study problems in mathematics and other scientific disciplines. Within mathematics these connections include links to algebraic geometry, representation theory, probability and topology. Links to other sciences include, for instance, quantum field theory, quantum electrodynamics, and superpotentials in physics. The aim of my research program is to further develop some of the interconnections between representation theory, combinatorics and probability. The mathematical notion of a representation developed, in part, from research on symmetries of physical objects. Information on the symmetries of the objects lead to information about the objects themselves. For example, the inherent symmetries of a crystal classify many of the physical properties of the crystals. This idea is reflected precisely in the mathematical theory of representations. Given an algebraic structure, one aims to classify all the objects on which it acts as symmetries; and determine the properties that are invariant under these symmetries. Broadly speaking, representation theory allows us to investigate the interactions of these algebraic structures with other parts of mathematics and with other scientific domains. The originality of my approach is to take advantage of the connections with probability theory, in particular with random walks on hyperplane arrangements (a basic combinatorial invariant associated with a set of symmetries). This approach has already had several successes and has opened the door to various new avenues of research. Some of these will facilitate the training of Masters and PhD students. Specific avenues of research vary in topic and include: investigating the behaviour of certain Markov chains; developing tools to study the representation theory of algebras; and unravelling the mysteries surrounding the new combinatorial Hopf algebra of supercharacters.