Elementary divisors of Specht modules

Let ℋq(Sn) be the Iwahori-Hecke algebra of the symmetric group defined over the ring Z[q,q−1]. The q-Specht modules of ℋq(Sn) come equipped with a natural bilinear form. In this paper we try to compute the elementary divisors of the Gram matrix of this form (which need not exist since Z[q,q−1] is not a principal ideal domain). When they are defined, we give the relationship between the elementary divisors of the Specht modules Sq(λ) and Sq(λ′), where λ′ is the conjugate partition. We also compute the elementary divisors when λ is a hook partition and give examples to show that in general elementary divisors do not exist.