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# Title: The root posets and their rich antichains

Abstract: 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 $5$.
 Comments: This is a completely revised version, now with reference to the exponents. The (n-1)-antichains are now called rich antichains and there is an outline in which way the root poset can be recovered from the set of rich antichains Subjects: Combinatorics (math.CO); Representation Theory (math.RT) Cite as: arXiv:1306.1593 [math.CO] (or arXiv:1306.1593v2 [math.CO] for this version)

## Submission history

From: Claus Michael Ringel [view email]
[v1] Fri, 7 Jun 2013 02:41:12 GMT (14kb)
[v2] Sat, 20 Jan 2018 07:52:28 GMT (28kb)
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