Parabolic subgroups of complex braid groups: the remaining case

Abstract

Recently, Marin and Gonz\'alez-Meneses introduced a class of ``parabolic'' subgroups for generalized braid groups associated to arbitrary complex reflection groups. Using notably Garside group structures on these generalized braid groups, they proved general results on parabolic subgroups for all cases but one. This last case is that of the complex braid group B(G31), which has no Garside group structure known so far, but instead a Garside groupoid structure. Using this Garside groupoid structure, we complete the results of Marin and Gonz\'alez-Meneses by proving their main theorems for the parabolic subgroups of the complex braid group B(G31).