d-Complete Posets Generalize Young Diagrams for the Jeu de Taquin Property

Abstract

The jeu de taquin process produced a standard Young tableau from a skew standard Young tableau by shifting its entries to the northwest. We generalize this process to posets: certain partial numberings of any poset are shifted upward. A poset is said to have the jeu de taquin property if the numberings resulting from this process do not depend upon certain choices made during the process. Young diagrams are the posets which underlie standard Young tableaux. These posets have the jeu de taquin property. d-Complete posets are posets which satisfy certain local structual conditions. They are mutual generalizations of Young diagrams, shifted Young diagrams, and rooted trees. We prove that all d-complete posets have the jeu de taquin property. The proof shows that each d-complete poset actually has the stronger "simultaneous" property; this may lead to an algebraic understanding of the main result. A partial converse is stated: "Non-overlapping" simultaneous posets are d-complete.