Canonical Systems of Basic Invariants for Unitary Reflection Groups

It is known that there exists a canonical system for every finite real reflection group. In a previous paper, the first and the third authors obtained an explicit formula for a canonical system. In this article, we first define canonical systems for the finite unitary reflection groups, and then prove their existence. Our proof does not depend on the classification of unitary reflection groups. Furthermore, we give an explicit formula for a canonical system for every unitary reflection group.

CHARACTERIZATION FOR THE MODULAR PARTY ALGEBRA

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.

On the centralizer algebra of the unitary reflection group G(m,p,n)

The imprimitive unitary reflection group G(m, p, n) acts on the vector space V =Cn naturally. The symmetric group Sk acts on ⊗kV by permuting the tensor product factors. We show that the algebra of all matrices on ⊗kV commuting with G(m, p, n) is generated by Sk and three other elements. This is a generalization of Jones’s results for the symmetric group case [J].