The BMM symmetrising trace conjecture for groups $G_4,\,G_5,\,G_6,\,G_7,\,G_8$

TL;DR

This paper proves the BMM symmetrising trace conjecture for certain exceptional complex reflection groups using computational algorithms and specific bases for their associated generic Hecke algebras.

Contribution

It provides the first proof of the conjecture for groups G4 through G8 by combining algorithms across multiple programming languages.

Findings

Confirmed the conjecture for G4, G5, G6, G7, G8

Developed algorithms in C++, SAGE, GAP3, Mathematica

Identified suitable bases for the Hecke algebras

Abstract

We prove the BMM symmetrising trace conjecture for the exceptional complex reflection groups $G_4,\,G_5,\,G_6,\,G_7,\,G_8$ using a combination of algorithms programmed in different languages (C++, SAGE, GAP3, Mathematica). Our proof depends on the choice of a suitable basis for the generic Hecke algebra associated with each group.