Iwahori - Hecke algebras of type A at roots of unityFrederick

Let H n be the Iwahori-Hecke algebra of type A n?1. For each Young diagram , an H n module S , called a Specht module, was deened in DJ]. The dimension of S does not depend on the choice of q, and for q not a root of unity, the S 's provide a complete set of irreducible H n-modules, up to isomorphism. If q is a primitive l-th root of unity, a complete set of simple H n-modules (D) has been constructed in DJ], where runs through all Young diagrams with at most l ? 1 rows of equal length. These modules are not well understood. The dimensions of the simple modules D can be computed if one knows the multiplicities d of the D in a composition series of S , for all possible 's and 's. A result known as Nakayama conjecture, or equivalently, in the context of quantum groups, as linkage principle, gives some information about the d 's. It says that d 6 = 0 only if + is in the orbit of + with respect to an aane reeection group. Here = (k ? 1; k ? 2; :::; 1; 0) for some k n, and the reeection group is that generated by the aane reeections in the hyperplanes y i ? y j = ml, 1 i < j k and m 2 Z. However, these orbits can become quite large and can contain many diagrams for which d = 0. In this paper, we explore the use of path idempotents for the Hecke algebra at roots of unity. The path idempotents are deened via the orthogonal representation, which is not well deened at roots of unity. Nevertheless, we show that certain sums of path idempotents are still well deened at roots of unity and can be used to derive information about simple modules and decomposition numbers at roots of unity. Our main technical result (Theorem 3.3.1) concerns a (non-unital) embedding of the (k-row quotient of) the Hecke algebra H m (x l) deened over the eld of rational functions K(x) into the (k-row quotient of the) Hecke algebra H n (x), for certain m and n. If q is a primitive l-th root of unity, this gives an embedding of KS m into H n (q). Using this, we can recover at least some special cases of irreducibility criteria of Dipper …