The reverse operator orbits on $\Delta(1)$ and a conjecture of Panyushev

TL;DR

This paper confirms Panyushev's conjecture regarding the structure of reverse operator orbits on certain posets associated with simple Lie algebra gradings, advancing understanding in algebraic combinatorics.

Contribution

The paper proves Panyushev's conjecture 5.11, providing new insights into the behavior of reverse operator orbits on weight posets.

Findings

Verification of Panyushev's conjecture 5.11

Enhanced understanding of reverse operator dynamics

Implications for algebraic combinatorics and Lie algebra theory

Abstract

We verify conjecture 5.11 of Panyushev [\emph{Antichains in weight posets associated with gradings of simple Lie algebras}, Math Z 281(3):1191--1214, 2015].