2003/07/31 by Uwe Grimm, Paul Martin, Paul P. Martin
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Random Matrices and Applications #hep-th #math-ph #math.MP #math.RA #msc:81R12 #msc:82B20 #msc:82B23
paper · pdf · doi:10.1088/0305-4470/36/42/010
published as J.Phys.A36:10551-10572,2003 · 24 pages, LaTeX using IOP styles, many figures included; typo in formula (7) corrected
openalex publication_date 2003/10/08 · arxiv created 2003/10/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
We define new diagram algebras providing a sequence of multiparameter generalizations of the Temperley–Lieb algebra, suitable for the modelling of dilute lattice systems of two-dimensional statistical mechanics. These algebras give a rigorous foundation to the various 'multi-colour algebras' of Grimm, Pearce and others. We determine the generic representation theory of the simplest of these algebras, and locate the nongeneric cases (at roots of unity of the corresponding parameters). We show by this example how the method used (Martin's general procedure for diagram algebras) may be applied to a wide variety of such algebras occurring in statistical mechanics. We demonstrate how these algebras may be used to solve the Yang–Baxter equations.