Abstract
Inspired by the infinite families of finite and affine root systems, we define a "stretching" operation on general crystallographic root systems which, on the level of Coxeter diagrams, replaces a vertex with a path of unlabeled edges. We embed a root system into its stretched versions using a similar operation on individual roots. For a fixed root, we describe the long-term behavior of two associated structures as we lengthen the stretched path: the downset in the root poset and Reading's arrangement of shards. We show that both eventually admit a uniform description, and deduce enumerative consequences: the size of the downset is eventually a polynomial, and the number of shards grows exponentially.
Highlights
In many questions about root systems, Coxeter groups, and related objects, the type A family is foundational, and often more resolved than the general case
Type A is foundational to the classification of finite and affine root systems: the type A Coxeter diagrams are paths, and almost1 every infinite family is described by Coxeter diagrams obtained by inserting paths into a fixed diagram
We generalize this kind of family to arbitrary Coxeter diagrams, using a stretching operation:
Summary
In many questions about root systems, Coxeter groups, and related objects, the type A family is foundational, and often more resolved than the general case. Type A is foundational to the classification of finite and affine root systems: the type A Coxeter diagrams are paths, and almost every infinite family is described by Coxeter diagrams obtained by inserting paths into a fixed diagram. We generalize this kind of family to arbitrary Coxeter diagrams, using a stretching operation:. Let G be a Coxeter diagram, x a vertex of G, and Lx Rx a partition of the neighbors of x into two subsets. In looking at a family of stretches of a diagram, two natural questions arise:. Hepworth [5] showed homological stability for Coxeter groups in stretched families with Rx = ∅. We examine the above questions in the context of combinatorial properties of roots
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