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For every composition of a positive integer, we construct a chain complex of modules whose terms are direct sums of tensor products of homomorphism spaces between modules over a Hecke algebra of the symmetric group on n letters. The construction is combinatorial and many counting arguments are employed. We conjecture that for every partition the chain complex has homology concentrated in one degree and that it is isomorphic to the dual of the Specht module. We prove the exactness in special cases. Along the way we visit many important results as it relates to the Hecke algebra of the symmetric group and permutation modules. It is a rich and deep area of study as it relates to the representation theory of the general linear group over a finite field.