The explicit construction of irreducible representations of the quantum algebras U_q(sl(n))

Burdik, C., King, R.C. and Welsh, T.A. (2001) The explicit construction of irreducible representations of the quantum algebras U_q(sl(n)). In, New Developments in Fundamental Interaction Theories: 37th Karpacz Winter School of Theoretical Physics. (AIP Conference Proceedings, 589) 37th Karpacz Winter School of Theoretical Physics (06/02/01 - 15/02/01) New York, USA. American Institute of Physics, pp. 158-169. (doi:10.1063/1.1419323).

Record type: Book Section

Abstract

The duality between the quantum algebra Uq(sl(n)) and the Hecke algebra Hm(q2) first pointed out by Jimbo is exploited to construct explicit irreducible representations of Uq(sl(n)). The method is based on the use of Young tableaux and involves the notion of q-dependent Young symmetrisers. A key role is played by q-dependent generalisations of the Garnir identities. The appropriate algorithm is first described and illustrated in the generic case for which q is not a root of unity. All matrix elements for the irreducible representations of Uq(sl(3)) are given. The complications that arise in the non-generic case for which q is a primitive p-th root of unity are then addressed. Explicit results on both irreducible and indecomposable representations are presented.

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