Elementary divisors of Specht modules

The irreducible representations of the symmetric groups and their Iwahori-Hecke algebras have been classified and constructed by James [6] and Dipper and James [2], yet simple properties of these modules, such as their dimensions, are still not known. Every irreducible representation of these algebras is constructed by quotienting out the radical of a bilinear form on a particular type of module, known as a Specht module. The bilinear forms on the Specht modules are the objects of our study. One way of determining the dimension of the simple modules would be to first find the elementary divisors of its Gram matrix over Z[q, q] and then specialize. This would also give the dimensions of the subquotients of the Jantzen filtrations of the Specht modules over an arbitrary field; see [7]. In general, such an approach is not possible because, as Andersen has shown, Gram matrices need not be diagonalizable over Z[q, q]; see [1, Remark 5.11]. We also give some examples of non–diagonalizable Specht modules in section 7. Let G(λ) be the Gram matrix of the Specht module S(λ). Then the first result in this paper shows that G(λ) is diagonalizable if and only if G(λ) is diagonalizable, where λ is the partition conjugate to λ. Moreover, ifG(λ) is divisibly diagonalizable (that is, G(λ) is equivalent to a diagonal matrix diag(d1, . . . , dm) such that di divides di+1, for 1 ≤ i < m), then so is G(λ ). In this case we can speak of elementary divisors and we show how the elementary divisors of G(λ) and G(λ) determine each other. This is a q–analogue of the corresponding result for the symmetric group [8]. We next consider the elementary divisors for the hook partitions. We show that when λ = (n − k, 1), for 0 ≤ k < n, the Gram matrix G(λ) is always divisibly diagonalizable over Z[q, q], and we determine the elementary divisors. Again, this is a q–analogue of the corresponding result for the symmetric groups [8], however, the proof in the Hecke algebra case is more involved and requires some interesting combinatorics.