Invariant Theory for Finite-Dimensional Algebras

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The PI proposes several projects in the representation theory of finite dimensional algebras that combine ideas and techniques from invariant theory, algebraic combinatorics, and algebraic geometry. One of the fundamental problems in this area is that of classifying the indecomposable representations. Based on the complexity of the indecomposable representations, a finite dimensional algebra is of tame or wild representation type. The PI will work on projects aimed at characterizing the representation type of an algebra in terms of the invariant theory of the algebra in question. This proposal deals first with developing reduction techniques for the study of quotient varieties of representations. The PI next proposes to use these techniques to characterize the tameness of an algebra in terms of the fields of rational invariants and the moduli spaces of representations of the algebra in question. The PI will also work on several other projects in quiver invariant theory and related problems about configurations of subspaces, polynomiality properties of tensor product multiplicities, and Mori dream spaces.Representation theory, a branch of modern algebra studying symmetries of various objects of interest, interacts with many other areas in mathematics, mathematical physics, chemistry, and theoretical computer science. The building blocks of the objects studied in the representation theory of algebras are the indecomposable representations. The projects in this proposal aim to provide geometric means for parametrizing the indecomposable representations, and to explore applications to problems in algebraic combinatorics and algebraic geometry.