Given a Cartan matrix related to a Coxeter diagram, I can modify it by changing one of many edges within the diagram with an extended chain of vertices related by merely laced edges; for instance, that is the place a lot of the infinite households of finite or affine diagrams come from.
Given a constructive root of the foundation system related to the unique Cartan matrix, we will write it within the foundation of easy roots, and thus (utilizing the affiliation of easy roots to vertices) as a operate on the vertices. I am keen on taking a root written on this approach and equally stretching it out to suit the prolonged Coxeter diagram by making the coefficients on the added vertices all the identical, like so:
Particularly, I am keen on what occurs to the hyperplane association of shards (within the sense of Nathan Studying, launched right here) of a root constructed this fashion because the chain will get longer. However earlier than trying into that, I might wish to know generally if something extra is known about how root techniques behave below this extension. So I’ve a obscure query and a selected query:
1) What earlier work has been executed on the habits and stability of those households of root techniques? I am undecided how precisely to go about looking for it; the one factor my advisor and I may discover was Homological stability for households of Coxeter teams, which fits in a unique course and would not seem to have related references.
2) In an try to slim down the (sometimes) infinitely many roots at play, suppose I prohibit consideration to roots (seen as features on the Coxeter diagram) with coefficients bounded by a specific quantity. How will the variety of such bounded roots develop because the graph lengthens? Some preliminary number-crunching suggests it grows polynomially, and my hunch could be that there are some easy restrictions on the coefficients such that counting all of the features satisfying these restrictions is asymptotically shut, however I am undecided what sort of restrictions are applicable.