HIGHEST WEIGHT VECTORS OF IRREDUCIBLE REPRESENTATIONS OF THE QUANTUM SUPERALGEBRA Uq (gl(m,n))

The Iwahori-Hecke algebra Hk(q ) 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 Hk(q ) 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 , for k = 1, 2, . . . . 0. Introduction One of the main studies of the representation theory of a semisimple Lie algebra g is constructing all the irreducible g-modules. Related with this problem, we are also interested in obtaining highest weight vectors of the irreducible summands of a tensor representation. When g is the special linear Lie algebra sl(n) or the general linear Lie algebra gl(n) over the field C, this problem was successfully solved by I. Schur in [13] and [14]. Schur investigated the tensor product space of the natural representation, which is the irreducible representation of gl(n) with highest weight ǫ1. He showed the action of gl(n) on the tensor product space generate the full centralizer of the symmetric group action. And then, from the double centralizer theorem, we may show the associative algebra generated by actions commuting with actions of gl(n), which is called the centralizer algebra of gl(n), is a quotient of the group algebra CSk of the symmetric group Sk. This result is often called Schur-Weyl duality, and it is important for understanding the representation theory of gl(n). Schur used results on the representation theory of the symmetric group Sk by F. Frobenius [5] and by A. Young [15]. Schur used the decomposition of the group algebra CSk to obtain the irreducible decomposition of the tensor product space via the Young symmetrizers. Same approach was made by A. Berele and A. Regev [3] and G.Benkart and C. Lee Shader [2] for the general linear Lie superalgebra gl(m,n). When g = gl(m,n), the centralizer algebra is again a homomorphic image of CSk, and we can also use the Young symmetrizers to decompose the tensor product space. In 1986, M. Jimbo [7] constructed the Drinfel’d-Jimbo quantum group Uq(gl(n)) of gl(n). He also showed the action of the Iwahori-Hecke algebra of Type A, Hk(q ), on the k-fold tensor product space of the natural representation commutes with the action of Uq(gl(n)). And a q-analogue of the Young symmetrizers was obtained by A. Gyoja [6]. The usual trick for proving that the action of general linear Lie algebra gl(n) generates the full centralizer of the symmetric group action uses the idempotent ∑ σ∈Sk σ to construct a projection map onto the gl(n)-invariants. Unfortunately this 1991 Mathematics Subject Classification. Primary 17B37,20C15; Secondary 17B70, 05A17. Research supported in part by National Science Foundation grant DMS-9622447. 1