In this paper, we describe a study into the explicit construction of irreducible representations of the Hecke algebra Hn(q) of type An-1 in the non-generic case where q is a root of unity. The approach is via the Specht modules of Hn(q) which are irreducible in the generic case, and possess a natural basis indexed by Young tableaux.
The general framework in which the irreducible non-generic Hn(q)-modules are to be constructed is set up and exploited in the case of two-part partitions. For such partitions, we obtain the composition series of the Specht modules, describe a basis for each irreducible module in terms of a subset of the set of standard tableaux, and detail an algorithm by which their explicit matrix representations may be calculated. Plentiful examples are given. Full proofs will be given elsewhere.
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