On orbits of antichains of positive roots

TL;DR

This paper investigates the properties of an operator acting on antichains of positive roots in finite posets, especially those related to root systems, and explores invariant functions and conjectural behaviors.

Contribution

It introduces and studies the operator on antichains for root system posets, providing new insights and conjectures for specific cases like type A_n.

Findings

Properties of the operator on antichains are established for certain root systems.

An X-invariant integer-valued function on antichains of Δ+ is constructed and analyzed.

Conjectural properties of the operator are discussed for graded posets associated with root systems.

Abstract

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 $\Delta^+$ is the set of positive roots and $\Pi$ is the set of simple roots in $\Delta^+$, then we consider the cases $P=\Delta^+$ and $\Delta^+\setminus \Pi$. For the root system of type $A_n$, we consider an $X$-invariant integer-valued function on the set of antichains of $\Delta^+$ and establish some properties of it.