Representations of quivers, their generalizations and invariants

箭袋的表示、概括和不变量

• 批准号: 76879601
• 负责人: Dr. Artem Lopatin
• 金额: --
• 依托单位: Institute of Mathematics
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2008
• 资助国家: 德国
• 起止时间: 2007-12-31 至 2010-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/76879601?language=en
• 关键词: Representations; quivers; their; generalizations; invariants

A quiver Q is a finite oriented graph. Its representation is a collection of vector spaces assigned to vertices together with linear mappings “along” arrows. To be able to consider bilinear forms on vertex vector spaces together with linear mappings between them the notion of mixed quiver settings was introduced. To define it we assume that vector spaces assigned to some vertices are dual to each other. All (mixed) representations with fixed dimensions of vector spaces constitute the space of representations H. The product of general linear groups G acts on vertex vector spaces as change of bases and thereby G acts on H. Orbits of this action correspond to isomorphic classes of representation of Q. Similarly, we can define the group G* for mixed representations. The first part of the research is dedicated to semisimple mixed representations and mixed representations with closed G*-orbits on H. The second part is dedicated to relations between semi-invariants, i.e., invariants with respect to the commutator of G, for the usual representations of quivers. We intend to apply this result to Littlewood-Richardson coefficients. The third part is dedicated to generators for quantum matrix invariants.

图Q是一个有限定向图。它的表示是分配给顶点的向量空间的集合以及“沿沿着”箭头的线性映射。为了能够考虑顶点向量空间上的双线性形式以及它们之间的线性映射，引入了混合双线性集的概念。为了定义它，我们假设分配给某些顶点的向量空间彼此对偶。向量空间的所有具有固定维数的（混合）表示构成表示空间H。一般线性群G的积作为基的变换作用在顶点向量空间上，从而G作用在H上。这个作用的轨道对应于Q的表示的同构类。类似地，我们可以为混合表示定义群G*。第一部分研究H上的半单混合表示和具有闭G*-轨道的混合表示。第二部分致力于半不变量之间的关系，即，不变量相对于换向器的G，通常表示的箭。我们打算将这一结果应用于Littlewood-Richardson系数。第三部分是量子矩阵不变量的生成器。