Let ${\mathcal H}_{q}(d)$ be the Iwahori–Hecke algebra of the symmetric group, where $q$ is a primitive $l$th 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 $l \geq 4$ the multiplicities in a Specht or dual Specht module filtration of an ${\mathcal H}_q(d)$-module are well defined. A cohomological criterion is given for when an ${\mathcal H}_{q}(d)$-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 $p \geq 5$. These results are false in general for $p = 2$ or $3$.