Stability of stretched root systems, root posets, and shards

TL;DR

This paper investigates how stretching root systems affects their combinatorial structures, revealing polynomial growth in downsets and exponential growth in shards, thus advancing understanding of root system stability.

Contribution

It introduces a stretching operation on root systems and analyzes its impact on root posets and shards, providing new uniform descriptions and growth behaviors.

Findings

Downsets grow polynomially with stretching.

Number of shards increases exponentially.

Both structures admit uniform descriptions after stretching.

Abstract

Inspired by the infinite families of finite and affine root systems, we consider 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 study the growth of two associated structures as we lengthen the stretched path: the downset in the root poset (in the sense of Bj\"orner and Brenti [3]) and the arrangement of shards, introduced by Nathan Reading. 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.