Minuscule representations and Panyushev conjectures

TL;DR

This paper proves four of Panyushev's conjectures related to root posets from Lie algebra gradings and explores their connections with minuscule representations and combinatorial phenomena.

Contribution

It provides proofs for four conjectures and links root posets to key algebraic and combinatorial identities and phenomena.

Findings

Proofs of four Panyushev conjectures.

Connections established between root posets and Kostant-Macdonald identity.

Linkage of posets with minuscule representations and cyclic sieving phenomenon.

Abstract

Recently, Panyushev raised five conjectures concerning the structure of certain root posets arising from $\mathbb{Z}$-gradings of simple Lie algebras. This paper aims to provide proofs for four of them. Our study also links these posets with Kostant-Macdonald identity, minuscule representations, Stembridge's "$t=-1$ phenomenon", and the cyclic sieving phenomenon due to Reiner, Stanton and White.