Geodesic Normal Forms and Hecke Algebras for the Complex Reflection Groups G(de,e,n)

TL;DR

This paper develops geodesic normal forms for complex reflection groups G(de,e,n), enabling explicit minimal word representatives and natural bases for associated Hecke algebras, leading to a new proof of the BMR freeness conjecture.

Contribution

It introduces a combinatorial technique to determine minimal word representatives and constructs natural bases for Hecke algebras of G(de,e,n).

Findings

Established geodesic normal forms for G(de,e,n)

Constructed natural bases for associated Hecke algebras

Provided a new proof of the BMR freeness conjecture

Abstract

We establish geodesic normal forms for the general series of complex reflection groups G(de,e,n) by using the presentations of Corran-Picantin and Corran-Lee-Lee of G(e,e,n) and G(de,e,n) for d > 1, respectively. This requires the elaboration of a combinatorial technique in order to explicitly determine minimal word representatives of the elements of G(de,e,n). Using these geodesic normal forms, we construct natural bases for the Hecke algebras associated with the complex reflection groups G(e,e,n) and G(d,1,n). As an application, we obtain a new proof of the BMR freeness conjecture for these groups.