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d-Complete Posets: Local Structural Axioms, Properties, and Equivalent Definitions

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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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Authors and Affiliations

  1. Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC, USA

    Robert A. Proctor & Lindsey M. Scoppetta

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  1. Robert A. Proctor
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  2. Lindsey M. Scoppetta
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Correspondence to Robert A. Proctor.

Additional information

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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