Coxeter n-cubes and Signed Groupoid-Sets

We investigate mathematical constructions that are related to a class of groupoids called generalized Brink-Howlett groupoids. Generalzied Brink-Howlett groupoids are groupoids that are built from the data of a Coxeter system. Generalized Brink-Howlett groupoids generalize the notion of regular Brink-Howlett groupoids, as introduced by Brink and Howlett.

We examine commutative diagrams in the shape of n-dimensional cubes whose edges are indexed by elements of a Coxeter system subject to certain constraints. Such commutative diagrams are called Coxeter n-cubes. Edges of Coxeter n-cubes can be viewed as morphisms of a generalized Brink-Howlett groupoid. We study some properties of Coxeter n-cubes in general. We enumerate the number of Coxeter n-cubes modulo reorientation in the rank n type A Coxeter system and describe the elements of the rank n type A Coxeter system that appear as some edge of a Coxeter n-cube.

We study how regular Brink-Howlett groupoids can be endowed with the structure of a signed groupoid-set. Signed groupoid sets were introduce by Dyer, Wang. A signed groupoid set is a groupoid with definitely involuted sets attached to each object of the groupoid. Definitely involuted sets share properties that are similar to root systems. Thus, signed groupoid sets generalize the notion of root systems to classes of groupoids. Using a particular signed groupoid set, we construct a generalization of the Tits cone and the imaginary cone to the class of regular Brink-Howlett groupoids.