Cluster combinatorics and Artin group presentations

Hugh Thomas

Associated to a finite reflection group W is a lattice of noncrossing partitions NC(W). It is of interest because it provides a Garside structure on the Artin group A corresponding to W other than the usual one. This "dual" structure has certain technical advantages. In particular, it has been instrumental in constructing some Eilenberg-Mac Lane spaces for A. >

If W is crystallographic, it was observed by Chapoton than the number of elements in NC(W) is the same as the number of clusters in the corresponding cluster algebra. A bijection was later given by Reading. The bijection was reinterpreted in representation-theoretic terms and extended to include affine type by Ingalls and the speaker. The goal of this talk is to discuss how this representation-theoretic approach can be productively applied to the study of Artin groups. I will give a uniform proof (for finite, crystallographic type) of the correctness of the presentation at the heart of the dual Garside structure for finite type Artin groups. (This was originally proved by Birman-Ko-Lee for type A, and by Bessis, on a type-by-type basis, for general finite type.) I will also explain the conjectural generalization of this result: a presentation for arbitrary Artin groups which is expressed in terms of exceptional sequences of quiver representations.