Pappe Joseph, Pfannerer-Mittas, S. H., Schilling, A., & Simone, M. C. (2023). Promotion for fans of Dyck paths. In Proceedings of the 35th Conference on Formal Power Series and Algebraic Combinatorics. 35th Conference on Formal Power Series and Algebraic Combinatorics (FPSAC’23), Davis, United States of America (the).
E104-06 - Forschungsbereich Konvexe und Diskrete Geometrie
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Published in:
Proceedings of the 35th Conference on Formal Power Series and Algebraic Combinatorics
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Volume:
89B
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Date (published):
2023
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Event name:
35th Conference on Formal Power Series and Algebraic Combinatorics (FPSAC'23)
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Event date:
17-Jul-2023 - 21-Jul-2023
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Event place:
Davis, United States of America (the)
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Number of Pages:
12
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Keywords:
crystal bases; promotion; Fomin growth and chord diagrams; Dyck paths
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Abstract:
We construct an injection from the set of r-fans of Dyck paths of length n into the set of chord diagrams on [n] that intertwines promotion and rotation. This is done in two different ways, namely as fillings of promotion matrices and in terms of Fomin growth diagrams. Our analysis uses the fact that r-fans of Dyck paths can be viewed as highest weight elements of weight zero in crystals of type Br, which in turn can be analyzed using virtual crystals. On the level of Fomin growth diagrams, the virtualization process corresponds to the Roby–Krattenthaler blow up construction. Our construction generalizes to vacillating tableaux as well. We give a cyclic sieving phenomenon on r-fans of Dyck paths using the promotion action.
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Project title:
Rotationsinvariante diagrammatische Basen für Invariantenräume: 25658 (Österr. Akademie der Wissenschaften)