In this paper, we describe a study into the explicit
construction of irreducible representations of the Hecke algebra
*H*_{n}(*q*) of type *A*_{n-1} in the non-generic case where *q* is
a root of unity. The approach is via the Specht modules of
*H*_{n}(*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
*H*_{n}(*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.

An extended version of that originally presented at the 4th International Colloquium "Quantum Groups and Integrable Systems," Prague, 22-24 June 1995; and appearing in Czech J. Phys.

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