In this paper, we give a characterization for the modular party algebra Pn,r(Q) by generators and relations. By specializing the parameter Q to a positive integer k, this algebra becomes the centralizer of the unitary reflection group G(r, 1, k) in the endomorphism ring of V⊗n under the condition that k ≥ n.

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Introduction. We construct an algebra H_I,M_I(a q) with complex parameters a and q. The centralizer algebra of a mixed tensor representation of CUq(gl) is a quotient of it. The HOMFLY polynomial of links in S is equal to a trace of H_,_l(a, q). Each irreducible character of it corresponds to an invariant of links in a solid torus. As an application, we get a formula for the HOMFLY polynomial of satellite links. The detail will be published elsewhere.1o The centralizer algebra of mixed tensor representation. The quantum group cUq(gl) is the q-analogue of the universal enveloping algebra CU(gl). The Lie algebra gl acts on V'--C naturally and it is called the vector representation. This representation can be deformed for the q-analogue CUq(gln) and is also called the vector representation. Let V* denote the dual representation of V. Since q(gl) is a Hopf algebra, it acts on V '') V V V V. N times M timesThis representation is called the mixed fensor representation of cq(gln). Let C') "= {x e End (V,))xa:ax for any a e q(gl)}.Then C '') is an algebra and is called the centralizer algebra with respect to V ''). Jimbo shows in [2] that C ,) is a quotient of the Iwahori-Hecke algebra H_l(q). Let q and a be generic complex parameters. In other words, they are not equal to 0 nor any root of unity. Let H_,_(a, q) be the algebra defined by the ollowing generators and relations.

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Among the unitary reflection groups, the one on the title is singled out by its importance in, for example, coding theory and number theory. In this paper we start with describing the irreducible representations of this group and then examine the semi-simple structure of the centralizer algebra in the tensor representation. IntroductionThe group, which we denote by H 1 , on the title of this paper consists of 96 matrices of size 2 by 2. It is the unitary group generated by reflections (u.g.g.r.), numbered as No.8 in . This group, as well as No.9 in the same list, has long been recognized. The purpose of the present paper is to give a contribution to H 1 by decomposing the centralizer algebra of H 1 in the tensor representation into irreducible components.We shall give an outline of the first statement in Abstract. The group H 1 naturally acts on the polynomial ring C[x, y] of 2 variables over the complex number field C, i.e.We consider the invariant ringThis ring has a rather simple structure. It is generated by two algebraically independent homogeneous polynomials of degrees 8 and 12, and conversely this nature characterizes the u.g.g.r. Broué-Enguehard [6] found a map