2018/12/10 by Georges Neaime, Neaime, Georges
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1812.03714
openalex publication_date 2018/12/10 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We define geodesic normal forms for the general series of complex reflection\ngroups G(de,e,n). This requires the elaboration of a combinatorial technique in\norder to determine minimal word representatives and to compute the length of\nthe elements of G(de,e,n) over some generating set. Using these geodesic normal\nforms, we construct intervals in G(e,e,n) that give rise to Garside groups.\nSome of these groups correspond to the complex braid group B(e,e,n). For the\nother Garside groups that appear, we study some of their properties and compute\ntheir second integral homology groups. Inspired by the geodesic normal forms,\nwe also define new presentations and new bases for the Hecke algebras\nassociated with the complex reflection groups G(e,e,n) and G(d,1,n) which lead\nto a new proof of the BMR (Brou 'e-Malle-Rouquier) freeness conjecture for\nthese two cases. Next, we define a BMW (Birman-Murakami-Wenzl) and Brauer\nalgebras for type (e,e,n). This enables us to construct explicit Krammer's\nrepresentations for some cases of the complex braid groups B(e,e,n). We\nconjecture that these representations are faithful. Finally, based on our\nheuristic computations, we propose a conjecture about the structure of the BMW\nalgebra.\n