Rainbow Boomerang Graphs

Abstract

We generalize the well known exchange property of Coxeter groups to the setting of edge-colored graphs. This work aims to unify and extend the results of our companion article, "odd Verma's theorem", which were originally established for basic Lie superalgebras, to the broader setting of regular symmetrizable Kac-Moody Lie superalgebras and Nichols algebras of diagonal type, via the theory of Weyl groupoids in the sense of Heckenberger and Yamane. In particular, we show that the exchange property of odd reflections arises as a special case of the exchange property of Weyl groupoids. To study the exchange property itself, we analyze a class of edge-colored graphs introduced here, called rainbow boomerang graphs, which form an independently natural family of combinatorial objects. We also elaborate on odd Verma theorem in the specific setting of Nichols algebras of diagonal type.