Topological actions of Temperley-Lieb algebras and Representation Stability

We consider the Temperley-Lieb algebras $\textrm{TL}_n(\delta)$ at $\delta = 1$. Since $\delta = 1$, 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.