论文标题
Temperley-Lieb代数和表示稳定性的拓扑作用
Topological actions of Temperley-Lieb algebras and Representation Stability
论文作者
论文摘要
我们认为temperley-lieb代数$ \ textrm {tl} _n（δ）$ at $δ= 1 $。由于$δ= 1 $，我们可以考虑乘法单体结构，并询问该单体在拓扑空间上的作用。鉴于对拓扑空间的单体作用，我们对每个同源组都有代数作用。本文的主要定理明确地推论了同源组的表示结构，该结构与我们的$ \ textrm {tl} _n $ -space相关的自然过滤。作为这种结果的必然，我们能够研究稳定现象。有一种自然的方法可以在$ \ textrm {tl} _n（1）$的上下文中定义表示稳定性，而过滤的存在使我们能够定义拓扑稳定性的概念。我们能够推断出$ \ textrm {tl} _n $ spaces的过滤稳定序列导致同源组的表示稳定序列。这可以将其视为陈述的类似物，即配置空间的同源性形成有限生成的$ \ textrm {fi} $ - 模块。
We consider the Temperley-Lieb algebras $\textrm{TL}_n(δ)$ at $δ= 1$. Since $δ= 1$, we can consider the multiplicative monoid structure and ask how this monoid acts on topological spaces. Given a monoid action on a topological space, we get an algebra action on each homology group. The main theorem of this paper explicitly deduces the representation structure of the homology groups in terms of a natural filtration associated with our $\textrm{TL}_n$-space. As a corollary of this result, we are able to study stability phenomena. There is a natural way to define representation stability in the context of $\textrm{TL}_n(1)$, and the presence of filtrations enables us to define a notion of topological stability. We are able to deduce that a filtration-stable sequence of $\textrm{TL}_n$-spaces results in representation-stable sequence of homology groups. This can be thought of as the analogue of the statement that the homology of configuration spaces forms a finitely generated $\textrm{FI}$-module.
