Abstract
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.
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Acknowledgements
We thank Soichi Okada, Alexander Kleshchev, and Arun Ram for their help on Section 12, and Michael Strayer and the referee for helpful remarks on the exposition.
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This paper is based upon an April 2015 UNC Chapel Hill Masters project by Lindsey M. Scoppetta that was cowritten with project advisor Robert A. Proctor.
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Proctor, R.A., Scoppetta, L.M. d-Complete Posets: Local Structural Axioms, Properties, and Equivalent Definitions. Order 36, 399–422 (2019). https://doi.org/10.1007/s11083-018-9473-4
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DOI: https://doi.org/10.1007/s11083-018-9473-4