Rigid reflections of rank 3 Coxeter groups and reduced roots of rank 2 Kac-Moody algebras (Q6090815)

In a recent paper by K.-H. Lee and K. Lee, rigid reflections are defined for any Coxeter group via non-self-intersecting curves on a Riemann surface with labeled curves. When the Coxeter group arises from an acyclic quiver, the rigid reflections are related to the rigid representations of the quiver. For a family of rank \(3\) Coxeter groups, it was conjectured in the same paper that there is a natural bijection from the set of reduced positive roots of a symmetric rank \(2\) Kac-Moody algebra onto the set of rigid reflections of the corresponding rank \(3\) Coxeter group. The authors prove this conjecture.