## 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
Full Text:

### References:

 [1] Cellini, P.; Papi, P., Ad-nilpotent ideals of a Borel subalgebra II, J. algebra, 258, 112-121, (2002) · Zbl 1033.17008 [2] Deza, M.; Fukuda, K., Loops of clutters, (), 72-92 [3] Fon-der-Flaass, D.G., Orbits of antichains in ranked posets, European J. combin., 14, 17-22, (1993) · Zbl 0777.06002 [4] Cameron, P.J.; Fon-der-Flaass, D.G., Orbits of antichains revisited, European J. combin., 16, 545-554, (1995) · Zbl 0831.06001 [5] Panyushev, D., Ad-nilpotent ideals of a Borel subalgebra: generators and duality, J. algebra, 274, 822-846, (2004) · Zbl 1067.17005 [6] Panyushev, D., Short antichains in root systems, semi-Catalan arrangements, and $$B$$-stable subspaces, European J. combin., 25, 93-112, (2004) · Zbl 1171.17301 [7] Panyushev, D., The poset of positive roots and its relatives, J. algebraic combin., 23, 79-101, (2006) · Zbl 1091.17008 [8] D. Panyushev, Two covering polynomials of a finite poset, with applications to root systems and ad-nilpotent ideals, Preprint MPIM 2005-9 = math.CO/0502386, 20 pp · Zbl 1295.06001 [9] Sommers, E., $$B$$-stable ideals in the nilradical of a Borel subalgebra, Canad. math. bull., 48, 460-472, (2005) · Zbl 1139.17303 [10] Винберг, Э.Б.; Онищик, А.Л., Семинар по группам ли и алгебраическим группам, (1988), Наука Москва, (in Russian). English translation: A.L. Onishchik, E.B.Vinberg, Lie Groups and Algebraic Groups, Springer, Berlin, 1990
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