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The Iwahori-Hecke algebra H k (q 2 ) of type A acts on tensor product space V ⊗k of the natural representation of the quantum superalgebra Uq(gl(m, n)). We show this action of H k (q 2 ) and the action of Uq(gl(m, n)) on the same space determine commuting actions of each other. Together with this result and Gyoja's q-analogue of the Young symmetrizer, we construct a highest weight vector of each irreducible summmand of the tensor product space V ⊗k , for k = 1, 2, . . . .

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We establish a connection between planar rook algebras and tensor representations V ⊗k of the natural two-dimensional representation V of the general linear Lie superalgebra gl(1|1). In particular, we show that the centralizer algebra End gl(1|1) (V ⊗k ) is the planar rook algebra CP k−1 for all k ≥ 1, and we exhibit an explicit decomposition of V ⊗k into irreducible gl(1|1)-modules. We obtain similar results for the quantum enveloping algebra U q (gl(1|1)) and its natural two-dimensional module V q .