Cellular algebras and diagram algebras in representation theory

Abstract

We discuss a circle of ideas for addressing problems in representation theory using the philosophy of cellular algebras, applied to algebras described in terms of diagrams. Cellular algebras are often generically semisimple, and have non-semisimple specialisations whose representation theory may be discussed by solving problems in linear algebra, which are formulated in the semisimple context, and are therefore tractable in some significant cases. This applies in particular to certain Temperley-Lieb quotients of Hecke algebras, both finite dimensional and affine, which may be described in terms of bases consisting of diagrams. This leads to the application of cellular algebra theory to an analysis of their representation theory, with corresponding consequences for the relevant Hecke algebras. A particular case is the determination of the decomposition numbers of some standard modules for the affine Hecke algebra of $GL_n$. These decomposition numbers are known (by Kazhdan-Lusztig) to be expressible in terms of the dimensions of the stalks of certain intersection cohomology sheaves, and we discuss how our results imply the rational smoothness of some varieties associated with quiver representations.