Stability of Stretched Root Systems, Root Posets and Shards
Stability of Stretched Root Systems, Root Posets and Shards
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DOI：
10.37236/10029
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发表时间：
2020-10
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影响因子：
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通讯作者：
Will Dana
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文献类型：
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作者：
Will Dana
受有限根系统和仿射根系统的无穷族的启发，我们在一般的结晶学根系统上定义了一种“拉伸”运算，它在Coxeter图的水平上用一条未标记的边的路径代替一个顶点。我们使用对单个根的类似操作将一个根系统嵌入到它的拉伸版本中。对于固定的根，我们描述了两个相关结构在延长延伸路径时的长期行为：根偏序集中的下集和Reding的碎片排列。我们证明了两者最终都接受统一的描述，并推导出计数结果：下集的大小最终是多项式的，碎片的数量呈指数增长。
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.