The principal indecomposable modules of the dilute Temperley-Lieb algebra
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Abstract
The Temperley-Lieb algebra \documentclass[12pt]{minimal}\begin{document}$\mathsf {TL}_{n}(\beta )$\end{document}TLn(β) can be defined as the set of rectangular diagrams with n points on each of their vertical sides, with all points joined pairwise by non-intersecting strings. The multiplication is then the concatenation of diagrams. The dilute Temperley-Lieb algebra \documentclass[12pt]{minimal}\begin{document}$\mathsf {dTL}_{n}(\beta )$\end{document}dTLn(β) has a similar diagrammatic definition where, now, points on the sides may remain free of strings. Like \documentclass[12pt]{minimal}\begin{document}$\mathsf {TL}_{n}$\end{document}TLn, the dilute \documentclass[12pt]{minimal}\begin{document}$\mathsf {dTL}_{n}$\end{document}dTLn depends on a parameter \documentclass[12pt]{minimal}\begin{document}$\beta \in \mathbb {C}$\end{document}β∈C, often given as β = q + q−1 for some \documentclass[12pt]{minimal}\begin{document}$q\in \mathbb {C}^\times$\end{document}q∈C×. In statistical physics, the algebra plays a central role in the study of dilute loop models. The paper is devoted to the construction of its principal indecomposable modules. Basic definitions and properties are first given: the dimension of \documentclass[12pt]{minimal}\begin{document}$\mathsf {dTL}_{n}$\end{document}dTLn, its break up into even and odd subalgebras and its filtration through n + 1 ideals. The standard modules \documentclass[12pt]{minimal}\begin{document}$\mathsf {S}_{n,k}$\end{document}Sn,k are then introduced and their behaviour under restriction and induction is described. A bilinear form, the Gram product, is used to identify their (unique) maximal submodule \documentclass[12pt]{minimal}\begin{document}$\mathsf {R}_{n,k}$\end{document}Rn,k which is then shown to be irreducible or trivial. It is then noted that \documentclass[12pt]{minimal}\begin{document}$\mathsf {dTL}_{n}$\end{document}dTLn is a cellular algebra. This fact allows for the identification of complete sets of non-isomorphic irreducible modules and projective indecomposable ones. The structure of \documentclass[12pt]{minimal}\begin{document}$\mathsf {dTL}_{n}$\end{document}dTLn as a left module over itself is then given for all values of the parameter q, that is, for both q generic and a root of unity.
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