A Combinatorial Generalisation of Rank two Complex Reflection Groups via Generators and Relations
Igor Haladjian
TL;DR
This work develops a combinatorial framework that realises rank-two complex reflection groups as a broad class of $J$-reflection groups $W_b^c(k,bn,cm)$, unifying toric and classical cases. It provides uniform presentations by generators and relations (Theorem Pres2) for these groups, aligning with the Broué–Malle–Rouquier (BMR) presentations when the groups are finite, and shows that the centers are cyclic. The paper also classifies $J$-reflection groups up to reflection isomorphisms and connects generalized reflections with complex reflections, thereby tying the $J$-group construction to the classical rank-two theory. Toric reflection groups appear as the $b=c=1$ specialization, situating the broader theory within Gobet’s toric framework, and the authors foreshadow an associated braid-group construction attached to each $J$-reflection group.
Abstract
Complex reflection groups of rank two are precisely the finite groups in the family of groups that we call J-reflection groups. These groups are particular cases of J-groups as defined by Achar & Aubert in 2008. The family of J-reflection groups generalises both complex reflection groups of rank two and toric reflection groups, a family of groups defined and studied by Gobet. We give uniform presentations by generators and relations of J-reflection groups, which coincide with the presentations given by Broué, Malle and Rouquier when the groups are finite. In particular, these presentations provide uniform presentations for complex reflection groups of rank two where the generators are reflections (however the proof uses the classification of irreducible complex reflection groups). Moreover, we show that the center of J-reflection groups is cyclic, generalising what happens for irreducible complex reflection groups of rank two and toric reflection groups. Finally, we classify J-reflection groups up to reflection isomorphisms.