This award supports six projects in invariant theory and representation theory. Four of these projects are focussed on questions in invariant theory, related to the dual pairs and the "method of multiplicity-free actions". They all are concerned with extending the range of examples in which one can effectively compute quantities of interest. Specifically, they are concerned with tensor products, weight spaces, plethysms, counting lattice points in certain polyhedra, generalized spherical harmonics, and multiplicity-one phenomena in restrictions of representations. A fifth project concerns the computation of explicit structure of principal series representations of real reductive groups. A sixth project involves the study of structure in congruence Hecke algebras on p-adic groups and its implications for the existence of minimal K-types in p-adic representation theory. The research support concerns the representation theory of finite groups. A group is an algebraic object used to study transformations. Because of this, group are a fundamental tool in physics, chemistry and computer science as well as mathematics. Representation theory is an important method foe determining the structure of groups.
