Diagram algebras, Hecke algebras and decomposition numbers at roots of unity

Abstract

We prove that the cell modules of the affine Temperley–Lieb algebra have the same composition factors, when regarded as modules for the affine Hecke algebra of type A, as certain standard modules which are defined homologically. En route, we relate these to the cell modules of the Temperley–Lieb algebra of type B, which provides a connection between Temperley–Lieb algebras on n and n−1 strings. Applications include the explicit determination of some decomposition numbers of standard modules at roots of unity, which in turn has implications for certain Kazhdan–Lusztig polynomials associated with nilpotent orbit closures. The methods involve the study of the relationships among several algebras defined by concatenation of braid-like diagrams and between these and Hecke algebras. Connections are made with earlier work of Bernstein–Zelevinsky on the “generic case” and of Jones on link invariants.