On orbits of antichains of positive roots
Abstract
Description
For any finite poset P, there is a natural operator $X$ acting on the antichains of P. We discuss conjectural properties of this operator for some graded posets associated with irreducible root systems. In particular, if $Δ^+$ is the set of positive roots and $Π$ is the set of simple roots in $Δ^+$, then we consider the cases $P=Δ^+$ and $Δ^+\setminus Π$. For the root system of type $A_n$, we consider an $X$-invariant integer-valued function on the set of antichains of $Δ^+$ and establish some properties of it.
12 pages, final version; to appear in Europ. J. Combinatorics
12 pages, final version; to appear in Europ. J. Combinatorics