Towers of recollement and bases for diagram algebras: planar diagrams and a little beyond

The recollement approach to the representation theory of sequences of algebras is extended to pass basis information directly through the globalisation functor. The method is hence adapted to treat sequences that are not necessarily towers by inclusion, such as symplectic blob algebras (diagram algebra quotients of the type-C̃ Hecke algebras). By carefully reviewing the diagram algebra construction, we find a new set of functors interrelating module categories of ordinary blob algebras (diagram algebra quotients of the type-B Hecke algebras) at different values of the algebra parameters. We show that these functors generalise to determine the structure of symplectic blob algebras, and hence of certain two-boundary Temperley-Lieb algebras arising in Statistical Mechanics. We identify the diagram basis with a cellular basis for each symplectic blob algebra, and prove that these algebras are quasihereditary over a field for almost all parameter choices, and generically semisimple. (That is, we give bases for all cell and standard modules.) Centre for Mathematical Science, City University, Northampton Square, London EC1V 0HB, UK Department of Mathematics, University of Colorado, Campus Box 395, Boulder, Colorado 80309-0395, USA Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK