Topological actions of Temperley–Lieb monoids and representation stability

Abstract

In this paper, we consider the Temperley–Lieb algebras [Formula: see text] at [Formula: see text]. Since [Formula: see text], 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 [Formula: see text]-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 [Formula: see text], 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 [Formula: see text]-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 FI-module.