A resolution of the Specht module over a Hecke algebra of the symmetric group

For every composition lambda of a positive integer r, we construct a finite chain complex of modules whose terms are direct sums of tensor products of homomorphism spaces between modules over the Hecke algebra of the symmetric group Sr. The construction is combinatorial and can be carried out over every integral domain R. We conjecture that for every partition the chain complex has homology concentrated in one degree (at the end of the complex) and that it is isomorphic to the dual of the Specht module. We prove the exactness in special cases.