Realization problems in Representation Theory and Algebraic Combinatorics

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I propose to study the representation theory of hypergroups. A hypergroup is a finite-dimensional associative algebra A with a distinguished basis B={b0, b1, , br-1} for which the multiplicative identity b0 = 1 lies in B, and B has the “pseudo-inverse” property: for every bi in B, there is a unique bi* in B for which the coefficient of b0 in bibi* is nonzero. So a hypergroup generalizes the familiar group concept with the group's inverse property replaced by the pseudo-inverse. *** My work will focus on how these structures can be represented as matrices over as small a field or ring as possible, dealing mainly with two types of hypergroup in addition to group algebras: adjacency algebras of association schemes, in which the nonidentity elements of the basis B can be identified with a collection of graphs, and integral table algebras, which are hypergroups in which the coefficient of every bk in a product of basis elements bibj is always a nonnegative integer. There is a hierarchy here: group algebras are adjacency algebras, and adjacency algebras are integral table algebras. Over the last 20 years, much of the representation theory of these kinds of hypergroups has been motivated by ideas from the representation theory of groups and algebras, and this has resulted in fruitful applications in areas such as graph theory, design theory, and coding theory. It has provided a framework for studies of modular data appearing in conformal field theory, and occasionally new ideas in group theory have been uncovered by those working out the algebraic properties of hypergroups. Representation theory of hypergroups is an emerging area of research in algebraic combinatorics internationally. Many of the new contributions are taking place in Asian nations, Europe, and the U.S., which makes it an area ripe with international collaborative and exchange opportunities for Canadians. *** There is a substantial computational algebra component to our approach, which mixes with skills and experience in ordinary and integral representation theory, group theory, ring theory, algebraic graph theory, and emerging ideas in algebraic combinatorics to produce a vibrant research and training environment. The main projects in this proposal are about finding descriptions of the smallest field of realization of irreducible representations of hypergroups, discovering techniques for constructing irreducible representations of hypergroups, describing the units of finite order that can be represented integrally in the basis of a noncommutative hypergroup, and determining the integral table algebras that can be realized as the adjacency algebra of an association scheme. Ongoing collaborative projects in the representation theory of groups concerning the Zassenhaus conjecture for integral group rings and on the multiplicity-free question for the Weil character of a unitary group of a finite local ring are also part of the proposal.