Specht Filtrations for Hecke Algebras of Type A

Abstract

Let Hq(d) be the Iwahori–Hecke algebra of the symmetric group, where q is a primitive 1th root of unity. Using results from the cohomology of quantum groups and recent results about the Schur functor and adjoint Schur functor, it is proved that, contrary to expectations, for l ⩾ 4 the multiplicities in a Specht or dual Specht module filtration of an Hq(d)-module are well defined. A cohomological criterion is given for when an Hq(d)-module has such a filtration. Finally, these results are used to give a new construction of Young modules that is analogous to the Donkin–Ringel construction of tilting modules. As a corollary, certain decomposition numbers can be equated with extensions between Specht modules. Setting q = 1, results are obtained for the symmetric group in characteristic p ⩾ 5. These results are false in general for p = 2 or 3.