Representation Theory and Combinatorics

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The proposed research problems of this proposal are in the field of combinatorial representation theory and have, as primary goals, combinatorial constructions of irreducible representations and their characters. The algebraic structures under study are all groups and algebras which come from, or are generalizations of, Lie theoretic objects. The primary funding request of this proposal is for support for three graduate students of the principal investigator. The graduate student research projects center on the combinatorial representation theory of (1) toroidal Lie algebras, toroidal quantum groups and toroidal Yangians, (2) complex reflection groups and (3) Yokonuma-Hecke algebras. The research projects of the proposer are focused on the the combinatorial representation theory of affine Hecke algebras and Iwahori-Hecke algebras. This research strongly uses both algebraic geometric and combinatorial tools.Most interesting symmetries are complex enough that it is difficult to study them directly and a representation is a way of extracting information about these symmetries. Thus is born ``representation theory''. However, very often the representations themselves are quite intricate and complicated and the goal of Combinatorial Representation Theory is to find elementary models which allow us to more easily determine properties of these representations: size, number of components, splitting and combination rules, and character. The type of models that are the most useful have the flavor of games for children, like LEGOS or ERECTOR sets, and yet these models enable one to obtain very explicit information about the fine structure of the corresponding representations. In analogy with molecular biology, a general representation is a large and complex structure like a molecule and is composed of smaller ``irreducible components'', analogous to atoms. The research in this proposal is directed towards determining the properties of the irreducible components and on determining which irreducible components appear in naturally occuring ``standard'' representations.