Combinatorics of reflection groups and clusters
Christian KrattenthalerThe past few years have seen the emergence of a new research branch at the interface of combinatorics, algebra and geometry: "Fuß-Catalan combinatorics," as some people call it. It has been observed that there are several sets of objects associated to reflection groups, such as the (generalised) non-crossing partitions of Armstrong, the generalised cluster complex of Fomin and Reading, certain regions in the extended Shi arrangement, the geometric chains of Athanasiadis, etc., which are all enumerated by the (Fuß)-Catalan numbers associated to the underlying reflection group. Why this is so is only partially understood. I shall give an introduction into this fascinating subject of algebraic combinatorics.