Representations of $\mathfrak{sl}( 2,\mathbb{C} )$ on Posets and the Sperner Property
Robert A. Proctor · SIAM Journal on Algebraic and Discrete Methods · 1982
A ranked partially ordered set is said to be Sperner if it has no antichain bigger than its largest rank. A necessary and sufficient condition for a ranked partially ordered set to be rank symmetric, rank unimodal and strongly Sperner is presented. This condition involves representations of $\mathfrak{sl} ( 2,\mathbb{C} )$. It is used to provide a new, short proof that this combination of properties is preserved under the product operation. The sufficient part of this condition is also used to provide new, simpler proofs that certain combinatorially interesting partially ordered sets are rank symmetric, rank unimodal and strongly Sperner.