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On orbits of antichains of positive roots. (English) Zbl 1165.06001
Let $$(P,\leq)$$ be a finite poset, $$\mathfrak{An}(P)$$ the set of all antichains of $$P$$. For $$\Gamma\in\mathfrak{An}(P)$$ the ideal $${\mathfrak I}(\Gamma)$$ is defined as $$\{x\in P\mid\exists y\in \Gamma$$ with $$y\leq x\}$$, and $${\mathcal X}(\Gamma)$$ is the set of maximal elements of $$P\setminus{\mathfrak I}(\Gamma)$$. The mapping $${\mathcal X}$$ is a permutation of the finite set $$\mathfrak{An}(P)$$, and if $$m$$ is the cardinality of $$\mathfrak{An}(P)$$, the cyclic subgroup of the symmetric group $$\Sigma_m$$, generated by $${\mathcal X}$$, is denoted by $$\langle{\mathcal X}\rangle$$. The orbits of $$\langle{\mathcal X}\rangle$$ in $$\mathfrak{An}(P)$$ are then discussed in several conjectures and theorems of this paper.
We cite from the summary: “We discuss conjectural properties of $${\mathcal X}$$ for some graded posets associated with irreducible root systems. In particular, if $$\Delta^+$$ is the set of positive roots and $$\Pi$$ is the set of simple roots in $$\Delta^+$$, then we consider the cases $$P=\Delta^+$$ and $$P= \Delta^+\setminus\Pi$$. For the root system of type $${\mathbf A}_n$$, we consider an $${\mathcal X}$$-invariant integer-valued function on the set of antichains of $$\Delta^+$$ and establish some properties of it.”

##### MSC:
 06A06 Partial orders, general 20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures 17B22 Root systems
##### Keywords:
antichains; orbits
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##### References:
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