Geodesic
Stability of stretched root systems, root posets and shards
Will Dana1
1University of Michigan
The electronic journal of combinatorics, Tome 28 (2021) no. 3
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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.
DOI : 10.37236/10029
Classification : 17B22, 06A11
Mots-clés : stretching, crystallographic root systems

Will Dana  1

1 University of Michigan 
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     author = {Will Dana},
     title = {Stability of stretched root systems, root posets and shards},
     journal = {The electronic journal of combinatorics},
     year = {2021},
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     doi = {10.37236/10029},
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Will Dana. Stability of stretched root systems, root posets and shards. The electronic journal of combinatorics, Tome 28 (2021) no. 3. doi: 10.37236/10029

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