Topological actions of Temperley-Lieb algebras and Representation Stability

Abstract

We consider the Temperley-Lieb algebras ${\text{TL}}_n(\delta )$ 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 ${\text{TL}}_n$-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 ${\text{TL}}_n(1)$, 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 ${\text{TL}}_n$-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 $\text{FI}$-module.

arXiv comment30 pages, 5 figures

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