Algebraic Combinatorics and Representation Theory

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My research program centres on interactions between representation theory, algebraic combinatorics and random walks on groups and semigroups.***The mathematical notion of a representation developed, in part, from investigations into the symmetries of physical objects. The symmetries of an object lead to information about the object itself. For example, if two crystals both exhibit the same intrinsic symmetries, then they will also exhibit the same physical properties. Here, we say that both crystals "represent" the symmetries.***This idea is reflected in the mathematical theory of representations. One investigates objects that manifest a given collection of symmetries. These are the representations of the symmetries. One understands them by decomposing them into smaller representations. The indecomposable representations are the building blocks from which all others representations can be constructed. This decomposition problem is very difficult in general.***A large portion of my research program is dedicated to developing new techniques for decomposing representations, and to applying them to longstanding decomposition problems: in probability theory (decomposing eigenspaces of symmetrized random walks); in group representation theory (decomposing higher Lie modules); in symmetric function theory (decomposing cohomology representations related to the Stanley-Stembridge e-positivity conjecture).***The anticipated outcomes include: advancing our general understanding of decomposition problems; improving our proficiency with decomposition problems by contributing new techniques; and making progress on these longstanding open problems.******One of the originalities of my approach exploits connections between algebra, combinatorics and probability theory. Specifically, I will work with algebraic objects related to random walks on hyperplane arrangements (a basic combinatorial invariant associated with a set of symmetries). This seminal theory was initiated by Bidigare, Hanlon and Rockmore (BHR) and further developed and extended by Brown and Diaconis. The above decomposition problems can be recast within this theory and my recent advances with Margolis and Steinberg allow the introduction of topological tools. This presents new promising approaches that have already had several successes.******My research program can be implemented in phases, and can easily integrate the training of Masters and PhD students. It can also accommodate undergraduate students, since some portions are very tractable yet stimulating. Furthermore, many facets of my program are amenable to computer algebra explorations. This provides several HQP benefits: the effective development of intuition for research problems; the ability to explore high level examples that would be hard to explore otherwise; and the development of proof strategies through high end formal algebra manipulations.*** **