The root posets and their rich antichains
The root posets and their rich antichains
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DOI：
10.1360/n012018-00015
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发表时间：
2013-06
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通讯作者：
C. Ringel
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作者：
C. Ringel
设$\Delta$为秩$n\ge 2$的（连通的）Dynkin图，$\Phi_+ = \Phi_+(\Delta)$为对应的根序集（它由相对于固定根基的所有正根组成）。$\Phi_+$的宽度为$n$。我们将证明$\Phi_+$是“圆锥”的：它是$n$固体链的不相交并。$\Phi_+$中的富反链是基数$n-1$的反链。众所周知，富反链的数目等于$\Phi_+$的基数。$\Phi_+$中富反链的集合$\mathcal R(\Delta)$本身可以看作是一个与$\Phi_+$非常相似但并不总是同构的偏序集。我们将证明始终存在唯一的富反链$A$，使得$A$生成的理想中包含任何富反链。对于$\Delta\neq \Bbb E_6$, $A$的所有根都具有相同的长度，即$e_2$，其中$e_1 \le e_2 \le \dots \le e_n$是$\Delta.$的指数。对于$\Delta = \Bbb E_6$，反链$A$由四个长度为$e_2 = 4$的根和一个长度为$5$的根组成。
Let $\Delta$ be a (connected) Dynkin diagram of rank $n\ge 2$ and $\Phi_+ = \Phi_+(\Delta)$ the corresponding root poset (it consists of all positive roots with respect to a fixed root basis). The width of $\Phi_+$ is $n$. We will show that $\Phi_+$ is "conical": it is the disjoint union of $n$ solid chains. The rich antichains in $\Phi_+$ are the antichains of cardinality $n-1$. It is well known that the number of rich antichains is equal to the cardinality of $\Phi_+$. The set $\mathcal R(\Delta)$ of rich antichains in $\Phi_+$ can itself be considered as a poset which is quite similar, but not always isomorphic, to $\Phi_+$. We will show that there always exists a unique rich antichain $A$ such that any rich antichain is contained in the ideal generated by $A$. For $\Delta\neq \Bbb E_6$ all roots in $A$ have the same length, namely $e_2$, where $e_1 \le e_2 \le \dots \le e_n$ are the exponents of $\Delta.$ For $\Delta = \Bbb E_6$, the antichain $A$ consists of four roots of length $e_2 = 4$ and one root of length $5$.