The principal indecomposable modules of the dilute Temperley-Lieb algebra

The Temperley-Lieb algebra \tln(\beta) 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 \dtl n(\beta) has a similar diagrammatic definition where, now, points on the sides may remain free of strings. Like \tl n, the dilute \dtl n depends on a parameter \beta\in\mathbb C, often given as \beta=q+q^{-1} for some q\in\mathbb C^\times. 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 \dtl n, its break up into even and odd subalgebras and its filtration through n+1 ideals. The standard modules \U{n,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 \dr{n,k} which is then shown to be irreducible or trivial. It is then noted that \dtl n 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 \dtl n 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.