We use simple diagrammatic techniques to analyse the ordinary representation theory of the Hecke algebras , and to construct modules (resp. representations) which are generically simple (irreducible) and well defined in every specialization of q, including roots of unity. We determine several physically important properties of these modules, generalizing properties of the Temperley - Lieb algebra and its diagrams which have proved useful for lattice models. We show how these results can be used to locate energy level crossings in invariant quantum spin chains, and locate a new crossing of the thermodynamic limit spin chain at as an example.