The root posets (and the hereditary abelian categories of Dynkin type)
Abstract:
Given a root system, the choice of a root basis divides the set of roots
into the positive and the negative ones, it also yields an ordering
on the set of positive roots. The set of positive roots with respect
to this ordering is called a root poset.
The root posets have attracted a lot of interest in the last years.
The set of antichains (with a suitable
ordering) in a root poset turns out to be a lattice, it is called
lattice of (generalized) non-crossing partitions.
As Ingalls and Thomas have shown,
this lattice is isomorphic to the lattice of thick subcategories
of a hereditary abelian category of Dynkin type.
The isomorphism can be used in order to provide conceptual proofs
of several intriguing counting results for non-crossing partitions.
Ringel
Last modified: Aug 18, 2013