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

TL;DR

This paper investigates the conditions under which a specific 2-dimensional representation of the Hecke algebra associated with the complex reflection group G_7 remains irreducible after specializing parameters to complex numbers.

Contribution

It provides a necessary and sufficient criterion for the irreducibility of the specialized representation of the Hecke algebra of G_7.

Findings

Derived a criterion for irreducibility after specialization

Identified parameter conditions ensuring irreducibility

Enhanced understanding of representations of complex reflection groups

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 $H(G_7, u)$.