Invariant Theory for Finite-Dimensional Algebras

PI在有限维代数的表示理论中提出了几个项目，这些项目结合了不变量理论、代数组合学和代数几何的思想和技术。这一领域的一个基本问题是对不可分解表示的分类。基于不可分解表示的复杂性，有限维代数可分为温和型或野型。PI将致力于根据所讨论的代数的不变理论来描述代数的表示类型的项目。这项建议首先涉及为研究表示的商种类而开发归约技术。PI下一步建议使用这些技术来根据有理不变量的域和所讨论的代数的表示的模空间来刻画代数的温和性。PI还将致力于箭图不变量理论的其他几个项目以及子空间的配置、张量积的多项式性质和Mori梦空间的相关问题。表示理论是现代代数的一个分支，研究各种感兴趣对象的对称性，与数学、数学物理、化学和理论计算机科学的许多其他领域相互作用。代数表示论所研究的对象的构件是不可分解的表示。该方案中的项目旨在为不可分解表示的参数化提供几何手段，并探索其在代数组合学和代数几何中的应用。

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.