Spin Chains as Modules over the Affine Temperley–Lieb Algebra

The affine Temperley–Lieb algebra aTLN(β) is an infinite-dimensional algebra over ℂ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {C}$\end{document} parametrized by a number β∈ℂ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\beta \in \mathbb {C}$\end{document} and an integer N∈ℕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N\in \mathbb {N}$\end{document}. It naturally acts on (ℂ2)⊗N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\mathbb {C}^{2})^{\otimes N}$\end{document} to produce a family of representations labeled by an additional parameter z∈ℂ×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$z\in \mathbb C^{\times }$\end{document}. The structure of these representations, which were first introduced by Pasquier and Saleur (Nucl. Phys., 330, 523 1990) in their study of spin chains, is here made explicit. They share their composition factors with the cellular aTLN(β)-modules of Graham and Lehrer (Enseign. Math., 44, 173 1998), but differ from the latter by the direction of about half of the arrows of their Loewy diagrams. The proof of this statement uses a morphism introduced by Morin-Duchesne and Saint-Aubin (J. Phys. A, 46, 285207 2013) as well as new maps that intertwine various aTLN(β)-actions on the periodic chain and generalize applications studied by Deguchi et al. (J. Stat. Phys., 102, 701 2001) and after by Morin-Duchesne and Saint-Aubin (J. Phys. A, 46, 494013 2013).