Topological actions of Temperley-Lieb algebras and Representation Stability
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arXiv
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866929296526802944 |
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| author | Sitaraman, Maithreya |
| author_facet | Sitaraman, Maithreya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2008_09636 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Topological actions of Temperley-Lieb algebras and Representation Stability Sitaraman, Maithreya Representation Theory High Energy Physics - Theory Mathematical Physics Algebraic Topology Rings and Algebras 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. |
| title | Topological actions of Temperley-Lieb algebras and Representation Stability |
| topic | Representation Theory High Energy Physics - Theory Mathematical Physics Algebraic Topology Rings and Algebras |
| url | https://arxiv.org/abs/2008.09636 |