The poset of positive roots and its relatives
The poset of positive roots and its relatives
复制标题
正根偏序集及其亲属
DOI：
10.1007/s10801-006-6030-9
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发表时间：
2005
影响因子：
0.8
通讯作者：
D. Panyushev
中科院分区：
文献类型：
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作者：
D. Panyushev
令 Δ 为具有正根子集 Δ+ 的根系。我们考虑一些与 Δ+ 相关的偏序集的哈斯图的边。对于每条边，自然地定义其类型，并且我们研究将边集划分为类型。对于 Δ+，类型是单根，对于 ad 幂零和阿贝尔理想的偏序集，类型是仿射单根。我们给出了给定类型的边集的几种描述以及边数的统一表达式。根据 Peterson 的结果，阿贝尔理想的数量为 2n，其中 Δ 的阶数。我们证明对应的哈斯图的边数为(n+1)2n−2。对于 Δ+ 和阿贝尔理想，我们计算每种类型的边数，并证明类型 α 的边数仅取决于根 α 的长度。
Let Δ be a root system with a subset of positive roots, Δ+. We consider edges of the Hasse diagrams of some posets associated with Δ+. For each edge one naturally defines itstype, and we study the partition of the set of edges into types. For Δ+, the type is a simple root, and for the posets ofad-nilpotent and Abelian ideals the type is an affine simple root. We give several descriptions of the set of edges of given type and uniform expressions for the number of edges. By a result of Peterson, the number of Abelian ideals is 2n, wherenis the rank of Δ. We prove that the number of edges of the corresponding Hasse diagram is (n+1)2n−2. For Δ+and the Abelian ideals, we compute the number of edges of each type and prove that the number of edges of type α depends only on the length of the root α.