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We introduce two order relations on finite Coxeter groups which refine theabsolute and the Bruhat order, and establish some of their main properties. Inparticular we study the restriction of these orders to noncrossing partitionsand show that the intervals for these orders can be enumerated in terms of thecluster complex. The properties of our orders permit to revisit several resultsin Coxeter combinatorics, such as the Chapoton triangles and how they arerelated, the enumeration of reflections with full support, the bijectionsbetween clusters and noncrossing partitions.

Noncrossing partitions, Bruhat order and the cluster complex