Interval Structures, Hecke Algebras, and Krammer's Representations for the Complex Braid Groups B(e,e,n)

TL;DR

This paper develops new algebraic and combinatorial tools for complex reflection groups and braid groups, introduces new algebraic structures, and constructs explicit representations, advancing understanding of their properties and conjectures.

Contribution

It introduces geodesic normal forms, new presentations for Hecke algebras, and constructs Krammer's representations for complex braid groups, providing new proofs and conjectures.

Findings

Constructed geodesic normal forms for G(de,e,n).

Defined new presentations and bases for Hecke algebras.

Constructed explicit Krammer's representations for certain complex braid groups.

Abstract

We define geodesic normal forms for the general series of complex reflection groups G(de,e,n). This requires the elaboration of a combinatorial technique in order to determine minimal word representatives and to compute the length of the elements of G(de,e,n) over some generating set. Using these geodesic normal forms, we construct intervals in G(e,e,n) that give rise to Garside groups. Some of these groups correspond to the complex braid group B(e,e,n). For the other Garside groups that appear, we study some of their properties and compute their second integral homology groups. Inspired by the geodesic normal forms, we also define new presentations and new bases for the Hecke algebras associated with the complex reflection groups G(e,e,n) and G(d,1,n) which lead to a new proof of the BMR (Brou\'e-Malle-Rouquier) freeness conjecture for these two cases. Next, we define a BMW (Birman-Murakami-Wenzl) and Brauer algebras for type (e,e,n). This enables us to construct explicit Krammer's representations for some cases of the complex braid groups B(e,e,n). We conjecture that these representations are faithful. Finally, based on our heuristic computations, we propose a conjecture about the structure of the BMW algebra.