Mathematical Sciences: Invariant Theory and Applications to Representation Theory

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Abstract Howe 9622916 The primary aspect of this project is to investigate aspects of invariant theory related to multiplicity- free actions, and actions on flag manifolds. Work of the proposer with J. Horvath describing isotropy groups of actions of GLn on products of flag manifolds associated to tame quivers will be used to study the occurrence of dense orbits in actions associated to wild quivers.Instances of the First Fundamental Theorem of Invariant Theory for some low dimensional reducible actions analyzable by means of multiplicity-free actions will be investigated. A major goal will be an investigation of certain rings important for understan- ding tensor products and related questions. It has been proved in some cases, and is suspected in others, that these rings may be free as modules over certain natural polynomial subrings. The description of nice bases for these free modules may provide insight into the Littlewood-Richardson Rule and related formulas. As time permits, investigations of the ordinary geometric import of geometric invariant theory, and of the Plancherel Theorem of p-adic groups via Hecke algebras, may also be undertaken. Expository Abstract: In mathematics, one is often interested in objects which may be different, but may be equivalent for some purposes. Thus, in elementary geometry there is the notion of congruence: two triangles, or other figures, are congruent if one may be moved so as to fit exactly over the other. In an industrial context, congruence is the idea underlying interchangeable parts. The idea of invariant theory is to provide numerical criteria for two objects to be equivalent, with respect to whatever is the relevant notion of equivalence. Thus, two triangles are congruent exactly when the lengths of their sides are the same triple of numbers. The numbers which may be used to determine whether or not two objects are equivalent are called *invariants(it)*. Invariant theory seeks to f ind invariants for geometric configurations. This project seeks to apply invariant theory in new directions. It will look for systems of invariants for some new classes of configurations, and it will also attempt to find some new structure in some of the most important systems of invariants. Symmetry will be heavily exploited, and systems whose structure is very strongly determined by symmetries will form the central objects of investigation.