Let H n q be the Iwahori-Hecke algebra of type A n−1 over a field K of characteristic 0. For each Young diagram λ, an H n q module S λ , called a Specht module, was defined in [DJ]. The dimension of S λ does not depend on the choice of q, and for q not a root of unity, the S λ 's provide a complete set of irreducible H n q -modules, up to isomorphism. If q is a primitive lth root of unity, a complete set of simple H n q -modules D µ has been constructed in [DJ], where µ runs through all Young diagrams with at most l − 1 rows of equal length. These modules are not well understood, but their dimensions can be computed if one knows the multiplicities d λµ of the D µ in a composition series of S λ , for all possible µ's and λ's. A result known as the Nakayama conjecture, or equivalently, in the context of quantum groups, as the linkage principle, gives some information about the d λµ 's. It says that d λµ = 0 only if µ + ρ is in the orbit of λ + ρ with respect to an affine reflection group. Here ρ = k − 1 k − 2 1 0 for some k ≥ n, and the reflection group is that generated by the affine reflections in the hyperplanes y i − y j = ml, 1 ≤ i < j ≤ k and m ∈ . * We thank the referee for an exceptionally careful reading of a previous version of this paper and for pointing the way towards several improvements in the exposition. H.