The orientifold Temperley--Lieb algebra

TL;DR

The work develops a graded representation-theoretic framework for two-boundary algebras by realizing them as finite-dimensional graded quotients of orientifold quiver Hecke algebras, establishing a graded cellular structure for the symplectic blob algebra, and deriving a conjectural LLT-style algorithm to compute graded decomposition numbers. The approach connects transfer-matrix algebras from statistical mechanics with categorification via orientifold quiver Hecke algebras, offering an explicit construction of graded simples in non-generic parameter regimes. A key outcome is the identification of a graded cellular basis for the orientifold TL algebra and a detailed combinatorial model using orientifold paths, ladder tableaux, and standard tableaux, which yields both structural results and computational tools. The paper moreover analyzes decomposition matrices across root-of-unity vs non-root-of-unity cases and demonstrates phenomena (e.g., Nakayama-type failures) that illuminate the rich block structure of these algebras and potential links to $ extit{imath}$quantum groups.

Abstract

We construct gradings on the simple modules of 2-boundary Temperley--Lieb algebras and symplectic blob algebras by realising the latter algebras as quotients of Varagnolo--Vasserot's orientifold quiver Hecke algebras. We prove that the symplectic blob algebras are graded cellular and provide a conjectural algorithm for calculating their graded decomposition matrices. In doing so, we give the first explicit family of finite-dimensional graded quotients of the orientifold quiver Hecke algebras, providing a new entry point for the structure of these algebras -- in the spirit of Libedinsky--Plaza's ``blob algebra approach'' to modular representation theory.

Figures (24)

• Figure 1: The nine possible two row shapes for $n=2$ (where $9=1+2^2+2^2$ for the choices of $j=1,2$).
• Figure 2: Example for the proof of \ref{['prop:calibrated B modules']} when $n=5$ and $i= \frac{5+1}{2} = 3$. Pictured below are the standard tableaux $\mathsf{t}$ of shape $\gamma = (\beta q^{-4}, \beta q^{-2}, \beta , \beta q^{2}, \beta q^{4})$. They are marked by which of $e_0, e_1, \dots, e_4$ act by $0$ on the corresponding weight vector $v_\mathsf{t}$ in $N(\gamma)$ (where $e_0$ acts by $0$ if and only if $\beta \in\{ \alpha_1,\alpha_2\}$). The tableau marked $\mathsf{t}_\star$ is the unique tableau for which $e_j v_{\mathsf{t}_\star} \ne 0$ for all $j = 1,\dots, 4$. The other tableaux marked by $\star$ are the remaining of the elements of $F_\mathrm{odd}$, those whose corresponding weight vectors are not annihilated by any odd $e_{2j-1}$. Below that list, find the orbits of $\mathsf{t}_\star$ under the action of $S_\mathrm{odd}= \langle s_1,s_3\rangle$ and of $S_\mathrm{even} = \langle s_2,s_4\rangle$.
• Figure 3: Example for the proof of \ref{['prop:calibrated B modules']} when $n=6$ and $i= \frac{6}{2} = 3$. Pictured below are the standard tableaux $\mathsf{t}$ of shape $\gamma = (\beta q^{-4}, \beta q^{-2}, \beta , \beta q^{2}, \beta q^{4}, \beta q ^6)$. They are marked by which of $e_0, e_1, \dots, e_5$ act by $0$ on the corresponding weight vector $v_\mathsf{t}$ in $N(\gamma)$ (where $e_0$ acts by $0$ if and only if $\beta \in\{ \alpha_1,\alpha_2\}$). The tableau marked $\mathsf{t}_\star$ is the unique tableau for which $e_j v_{\mathsf{t}_\star} \ne 0$ for all $j = 2,3,\dots, 5$. The other tableaux marked by $\star$ are the remaining of the elements of $F_\mathrm{even}$, those whose corresponding weight vectors are not annihilated by any even $e_{2j}$. Below that list, find the orbits of $\mathsf{t}_\star$ under the action of $S_\mathrm{odd}= \langle s_3,s_5\rangle$ and of $S_\mathrm{even} = \langle s_2,s_4\rangle$.
• Figure 4: A standard tableau of shape $(k,\beta)$ for $k=3$ and $n=19$.
• Figure 5: The tableau $\mathsf{t}_{(k,\beta)}$ for $k=3$ and $n=19$. We have that $\mathrm{res}(\mathsf{t}_{(k,\beta)})=(\beta,\,\beta^{-1}q^{2},\,\beta q^2,\, \beta^{-1}q^{4} ,\, \beta q^{4},\,\beta^{-1}q^{6},\,\beta q ^6,\,\dots)$.

Theorems & Definitions (65)

• Proposition 2.1: GN
• Lemma 2.2
• proof
• Definition 2.3: GN
• Remark 2.4
• Remark 3.1
• Definition 3.2
• Example 3.3
• Theorem 3.4: DR25a, DR25b
• Proposition 3.5