Minuscule representations and Panyushev conjectures Dedicated to the Memory of Professor Bertram Kostant

Recently, Panyushev(2015) raised five conjectures concerning the structure of certain root posets arising from 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 et al.(2004).