Elementary divisors of Specht modules

Let ℋ q ( S n ) be the Iwahori-Hecke algebra of the symmetric group defined over the ring Z [ q , q − 1 ] . The q -Specht modules of ℋ q ( S n ) 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 S q ( λ ) and S q ( 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.