Representations of quivers, their generalizations and invariants

一个颤振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”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及其之间的线性映射，引入了混合颤振设置的概念。为了定义它，我们假设分配给一些顶点的向量空间是对偶的。所有固定维向量空间的（混合）表示构成了表示h的空间。一般线性群G的积作为基的变换作用于顶点向量空间，从而G作用于h。这个作用的轨道对应于q的同构表示类。同样，我们可以定义混合表示的群G*。第一部分研究了h上的半简单混合表示和闭G*轨道的混合表示，第二部分研究了颤振通常表示的半不变量之间的关系，即关于G的对易子的不变量。我们打算将这个结果应用于Littlewood-Richardson系数。第三部分是量子矩阵不变量的生成。

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.