On the irreducibility of the complex specialization of the representation of the Hecke algebra of the complex re ection group G7

We consider a 2-dimensional representation of the Hecke algebra H(G7; u), where G7 is the complex re ection group and u is the set of indeterminates u = (x1; x2; y1; y2; y3; z1; z2; z3): After specializing the indetrminates to non zero complex numbers, we then determine a nec- essary and sucient condition that guarantees the irreducibility of the complex specialization of the representation of the Hecke algebra H(G7; u).