论文标题
拉伸根系，根本和碎片的稳定性
Stability of stretched root systems, root posets, and shards
论文作者
论文摘要
受到有限和仿射根系的无限家族的启发，我们考虑了一般晶体学根系的“拉伸”操作，在Coxeter图的级别上，它用未标记边缘的路径代替了顶点。我们使用单个根部的类似操作将根系嵌入其拉伸版本中。对于固定的根，我们研究了两个相关结构的生长，因为我们延长了拉伸路径：根部poset中的降低（在Björner和Brenti [3]的意义上）和碎片的排列，由Nathan Reading引入。我们表明，两者最终都接受了统一的描述，并推断出枚举后果：降低的大小最终是多项式，碎片数量呈指数增长。
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 Björner 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.
