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，这些项目将不变量理论、代数组合学和代数几何的思想和技术联合收割机结合起来。这一领域的基本问题之一是对不可分解的表示进行分类。基于不可分解表示的复杂性，有限维代数是驯服或野生表示类型。PI将致力于旨在根据代数的不变理论来表征代数的表示类型的项目。这个建议首先涉及发展约简技术，用于研究表示的商品种。PI接下来提出使用这些技术来表征代数在有理不变量的域和代数表示的模空间方面的驯服性。PI还将致力于其他几个项目的不变量理论和相关问题的配置的子空间，多项式性质的张量积的多重性，和森dream spaces.Representation理论，一个分支现代代数研究对称性的各种感兴趣的对象，与许多其他领域的数学，数学物理，化学和理论计算机科学的相互作用。代数表示论中研究对象的构造块是不可分解表示。本计划的目的是提供几何方法来参数化不可分解的表示，并探索在代数组合学和代数几何问题中的应用。

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.