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\(d\)-complete posets: local structural axioms, properties, and equivalent definitions. (English) Zbl 07143294
Summary: Although \(d\)-complete posets arose along the interface between algebraic combinatorics and Lie theory, they are defined using only requirements on their local structure. These posets are a mutual generalization of rooted trees, shapes, and shifted shapes. They possess Stanley’s hook product property for their \(P\)-partition generating functions and Schützenberger’s well-defined jeu de taquin rectification property. The original definition of \(d\)-complete poset was lengthy, but more succinct definitions were later developed. Here several definitions are shown to be equivalent. The basic properties of \(d\)-complete posets are summarized. Background and a partial bibliography for these posets is given.

MSC:
06 Order, lattices, ordered algebraic structures
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