On the Irreducibility of the Complex Specialization of the Representation of The Hecke Algebra of the Complex Reflection Group $G_7$

Abstract

We consider a 2-dimensional representation of the Hecke algebra $\mathcal{H}(G_7,u)$, where $G_7$ is the complex reflection group and $u$ is the set of indeterminates $u=(x_1,x_2,y_1,y_2,y_3,z_1,z_2,z_3)$. After specializing the indetrminates to non zero complex numbers, we then determine a necessary and sufficient condition that guarantees the irreducibility of the complex specialization of the representation of the Hecke algebra $H(G_7,u)$.