Resumen de Principal G-cone for a tree
For any tree G, we introduce G-cones consisting of chambers and enumerate the number of chambers contained in two particular (called principal) G-cones. The problem is equivalent to the combinatorial problem of the enumeration of linear extensions of two bipartite orderings on a tree G. We characterize the principal G-cones among other G-cones by the strict maximality of the number of their chambers, and give a formula for this maximal (called principal) number by a finite sum of hook length formulae. We explain the formula through the simplicial block decomposition of principal G-cones. The results have their origin and application in the study of the topology related to Coxeter groups and Artin groups.
