THE MODULAR TEMPERLEY-LIEB ALGEBRA

We investigate the representation theory of the Temperley-Lieb algebra, TLn(delta), defined over a field of positive characteristic. The principle question we seek to answer is the multiplicity of simple modules in cell modules for TLn over arbitrary fields. This provides us with the decomposition numbers for this algebra, as well as the dimensions of all simple modules. We obtain these results from purely diagrammatic principles, without appealing to realisations of TLn as endomorphism algebras of Uq(sl2) modules. Our results strictly generalise the known characteristic zero theory of the Temperley-Lieb algebras.