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.”


06A06 Partial orders, general
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
17B22 Root systems


antichains; orbits
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