I'm working on Root Systems (https://en.wikipedia.org/wiki/Root_system). I learnt the basic definitions of a basis, of the Weyl group, and I read the proof of the existence of such bases for a given root system.
Now, I'd like to get deeper in the topic : I read somewhere (https://arxiv.org/pdf/1112.0381.pdf - Corollary 5.14) that the Catalan numbers count the number of non-decreasing distinguished bases of the root systmem $A_n$.
This is for a school project I'm making, and my teacher told me that this was somehow linked to a group action of the braid group over the bases, or something of the sort...
Do you have any definition of a distinguished base ? Of a non-decreasing distinguished base ? By any chance, would you even have some litterature covering the link between the Catalan numbers and those bases ?
Thanks already !