Interval Garside structures for the complex braid groups 𝐵(𝑒,𝑒,𝑛)

Abstract

We define geodesic normal forms for the general series of complex reflection groups G ( e , e , n ) G(e,e,n) . This requires the elaboration of a combinatorial technique in order to explicitly determine minimal word representatives of the elements of G ( e , e , n ) G(e,e,n) over the generating set of the presentation of Corran–Picantin. Using these geodesic normal forms, we construct intervals in G ( e , e , n ) G(e,e,n) that are proven to be lattices. This gives rise to interval Garside groups. We determine which of these groups are isomorphic to the complex braid group B ( e , e , n ) B(e,e,n) and get a complete classification. For the other Garside groups that appear in our construction, we provide some of their properties and compute their second integral homology groups in order to understand these new structures.