Séminaire Lotharingien de Combinatoire, B35e (1995), 20 pp.

T. A. Welsh

Two-rowed A-type Hecke Algebra Representations at Roots of Unity

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.

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. 45 (1996), 283-291.

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