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Topological actions of Temperley-Lieb algebras and Representation Stability

Maithreya Sitaraman

Abstract

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.

Topological actions of Temperley-Lieb algebras and Representation Stability

Abstract

We consider the Temperley-Lieb algebras at . Since , 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 -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 , 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 -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 -module.

Paper Structure

This paper contains 25 sections, 18 theorems, 72 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Quick review: Temperley-Lieb algebras and their representation theory
  3. The definition of Temperley-Lieb algebras
  4. Link state representations of Temperley-Lieb algebras
  5. Describing topological actions of Temperley-Lieb algebras
  6. Topological translation of the Temperley-Lieb relations
  7. Any neighbor intersection equals the full intersection
  8. Long-distance intersections of the same cardinality are isomorphic
  9. Surjective actions and the filtration of retracts
  10. Representation stability for Temperley-Lieb algebras at $\delta = 1$
  11. Defining representation stability
  12. General form of a stable module criterion
  13. A chain of standard representations of $\mathop{\mathrm{TL}}\nolimits_n$ is representation stable
  14. Example: an infinite-link-state representation of $\mathop{\mathrm{TL}}\nolimits(\infty)$ which is not stable
  15. Homology groups as $\mathop{\mathrm{TL}}\nolimits_n$-representations
  16. ...and 10 more sections

Key Result

Lemma 3.1

Each $u_i$ maps to a retraction map.

Figures (5)

  • Figure 1: A schematic of the cycle map
  • Figure 2: We define the action of $\mathop{\mathrm{TL}}\nolimits_n$ on $X$ as follows: $r_1,r_2,r_3,r_4,r_5$ will be retractions onto $A_1,A_2,A_3,A_4,A_5$ in such a way that if $|i - j| \ge 2$, then $r_j(A_i)$ is a point and if $|i-j| = 1$, then $r_i(A_j) = A_i$.
  • Figure 3: An illustration of X. We will describe the maps $u_1,u_2,u_3,u_4$ below, which will make $X$ into a $\mathop{\mathrm{TL}}\nolimits_5$-space, with $A_i = Q \vee S_i$ for each $i$ as described above. Note that $H_1(X)^{\mathop{\mathrm{TL}}\nolimits_n} =H_1(Q) = H_1(S^1) \not= 0$
  • Figure 4: Above is a schematic of $X_{5,2}$. This has a filtration $S^2 \vee S^2 \supseteq S^2$. In particular, note that each $A_i \cong S^2 \vee S^2$. The numbers next to the copies of $S^2$ indicate the minimal intersection containing that copy. For example, $1,4$ means that the minimal intersection containing that copy is $A_1 \cap A_4$. The action is as follows: each $u_i$ preserves $A_i$, takes $A_{i-1}$ and $A_{i+1}$ isomorphically to $A_i$ and retracts $A_j$ to $A_i$ for $|j-i| \ge 2$. However, these isomorphisms and retractions must be chosen carefully so as to not lead to any contradictions - this is really the heart of the matter. In red arrows above, we have drawn the action of $u_1$. In blue arrows above, we have drawn the action of $u_2$. The action of $u_3$ is similar to the action of $u_2$ and the action of $u_4$ is similar to the action of $u_1$.
  • Figure 5: A schematic of $X_4$. We have taken $c_4 = 3$. The cubes of course depict $3$-tori. The numbers $1,2,3$ depict the retracts $A_1, A_2, A_3$ respectively - each of which is a $3$-torus. The red arrows depict the action of $u_1$ - $A_3$ retracts to the copy of $S^1 \subset A_1 \cap A_3$ while $A_2$ is taken isomorphically to $A_1$. The blue arrows depict the action of $u_2$ - both $A_1$ and $A_3$ are taken isomorphically to $A_2$. The action of $u_3$ is similar to the action of $u_1$.

Theorems & Definitions (67)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Lemma 3.1: Translating the idempotent relation
  • proof
  • Lemma 3.2: Translating the neighbor relation
  • proof
  • Remark 3.3
  • proof
  • Lemma 3.5: Translating the long-distance relation
  • ...and 57 more