论文标题
派生的字符图
Derived Character Maps of Groups Representations
论文作者
论文摘要
在本文中，我们构建和研究了$ \ infty $ groups的有限维表示的派生特征图。作为$ \ infty $群体的模型，我们采用同型简单基团，即群体理论上的同型简单代数（从Badzioch的意义上）。我们在此类组上定义了“组代数”的环状，对称和表示同源，并构建了与这些同源性理论相关的规范痕量图。在一维表示的情况下，我们表明我们的痕量图是拓扑起源：它们是由（迭代）环空间的自然图引起的，这些图在同型中的理论中得到了很好的研究。使用这种拓扑解释，我们推断出关于表示同源性的一些代数结果：特别是，我们证明了组代数的对称同源性和一维表示同源性自然是同构，前提是基本环$ k $是特征性零的领域。我们还研究了以$ n \ to \ infty $的稳定极限为$ n $维表示的派生字符图的行为，在这种情况下，我们表明它们“融合”成为同构。
In this paper, we construct and study derived character maps of finite-dimensional representations of $\infty$-groups. As models for $\infty$-groups we take homotopy simplicial groups, i.e. homotopy simplicial algebras over the algebraic theory of groups (in the sense of Badzioch). We define cyclic, symmetric and representation homology for `group algebras' over such groups and construct canonical trace maps relating these homology theories. In the case of one-dimensional representations, we show that our trace maps are of topological origin: they are induced by natural maps of (iterated) loop spaces that are well studied in homotopy theory. Using this topological interpretation, we deduce some algebraic results about representation homology: in particular, we prove that the symmetric homology of group algebras and one-dimensional representation homology are naturally isomorphic, provided the base ring $k$ is a field of characteristic zero. We also study the behavior of the derived character maps of $n$-dimensional representations in the stable limit as $ n\to \infty$, in which case we show that they `converge' to become isomorphisms.
