Minuscule representations and Panyushev conjectures

Abstract

Recently, Panyushev 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, Stanton and White.

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