Non-crossing partitions for exceptional hereditary curves

TL;DR

This paper introduces reflection groups of canonical type linked to hereditary curves and categorifies their non-crossing partitions using thick subcategories of coherent sheaves, also proving Hurwitz action transitivity.

Contribution

It provides a novel categorification of non-crossing partitions for new reflection groups and proves Hurwitz action transitivity in this context.

Findings

Categorification of non-crossing partitions for reflection groups of canonical type

Proof of Hurwitz action transitivity in these groups

Connection between reflection groups and hereditary curve categories

Abstract

We introduce a new class of reflection groups associated with the canonical bilinear lattices of Lenzing, which we call reflection groups of canonical type. The main result of this work is a categorification of the corresponding poset of non-crossing partitions for any such group, realized via the poset of thick subcategories of the category of coherent sheaves on an exceptional hereditary curve generated by an exceptional sequence. A second principal result, essential for the categorification, is a proof of the transitivity of the Hurwitz action in these reflection groups.