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

## Author

**John J. Graham** and **Gus I. Lehrer**

## Status

Research Report 2001-15

Date: 6 December 2001

## 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. 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.

## Key phrases

representations. braids. affine Hecke algebras. root of unity. diagram algebras.

## AMS Subject Classification (1991)

Primary: 20C08

Secondary: 16G30, 20G05, 20C35

## Content

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