论文标题
覆盖空间及其HOPF代数的不变性
Invariants that are covering spaces and their Hopf algebras
论文作者
论文摘要
在作者的先前论文中，构建了给定类型的代数结构的通用环。该环是由某些痕量图生成的多项式代数。结果表明，该戒指承认了一个理性的积极自我伴侣Hopf代数（缩写为理性的PSH-Elgebra），并猜想它始终接收一个晶格，该晶格是PSH-Elgebra，这是由Zelevinsky引入的结构。在本文中，我们回答了这一猜想，表明不变的通用环将有理psh-elgebras的张量分裂为单个变量中的多项式代数，或者接纳晶格是PSH-Elgebra。我们通过将图作为拓扑空间以及使用覆盖空间理论的工具来做到这一点。作为应用程序，我们得出了一个公式，该公式将Kronecker系数与自由组的有限索引亚组连接起来，以及其Weyl组的表示，以及有限生成的组中有限索引亚组的共轭类别数量的公式，该组将接受向整数群体进行表现同源的组合。
In a previous paper by the author a universal ring of invariants for algebraic structures of a given type was constructed. This ring is a polynomial algebra that is generated by certain trace diagrams. It was shown that this ring admits the structure of a rational positive self adjoint Hopf algebra (abbreviated rational PSH-algebra), and was conjectured that it always admits a lattice that is a PSH-algebra, a structure that was introduced by Zelevinsky. In this paper we answer this conjecture, showing that the universal ring of invariants splits as the tensor product of rational PSH-algebras that are either polynomial algebras in a single variable, or admit a lattice that is a PSH-algebra. We do so by considering diagrams as topological spaces, and using tools from the theory of covering spaces. As an application we derive a formula that connects Kronecker coefficients with finite index subgroups of free groups and representations of their Weyl groups, and a formula for the number of conjugacy classes of finite index subgroup in a finitely generated group that admits a surjective homomorphism onto the group of integers.
