Clusters in Lie theory

To continue on from where my previous post left off, many objects in Lie theory naturally carry the structure of a cluster algebra. Indeed, for many people – including Fomin and Zelevinsky – this was a substantial part of the motivation for the arcane definitions of the theory. The prototypical collection of cluster variables is the ‘flag minors’ on the base affine space of \text{SL}_n(\mathbb{C}) that obey generalised Pluecker relations, which are the archetypal exchange relations.

This next part outlines the wonderfully aesthetic cluster structure on base affine spaces, as well as starting to tie the cluster variables to the indecomposable modules over the preprojective algebra of the same Dynkin type as the group, so that the coincidence of finitude between these two objects in types A_n for n\leq 4 that I mentioned last time can be explained geometrically.

To prepare this post, I read the notes from Fomin’s ICM address in 2010. It was probably one of the most pleasant articles that I’ve read for a while; I highly recommend it. it contains some interesting historical commentary too that I wasn’t very familiar with. I come in with largely a mirror symmetric interest in cluster algebras, but the original motivation – asides from the compelling examples – was to study ‘positive varieties’. The basic idea is that, given a complex variety X equipped with some ‘natural’ set of functions \mathbf{\Delta}, one considers the positive variety

X_+:=\{x\in X:\Delta(x)\geq0\text{ for all }\Delta\in\mathbf{\Delta}\}.

This is often some sort of cone, for example in the case that X=N is a maximal unipotent subgroup of a simple Lie group of type ADE and \mathbf{\Delta} is the set of flag minors. The positive variety can contain a bizarre amount of information about the original variety: in the previous example, the strata in the cone know about the corresponding Bruhat order on Schubert cells. In their study of these objects, Fomin and Zelevinsky realised that a good source of ‘natural’ functions was cluster variables, and that hence the varieties to consider are cluster varieties.

Anyway, in the final part of this trilogy on clusters and semicanonical bases in Lie theory the two sides will be reconciled, allowing cluster techniques to help compute semicanonical basis elements.


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