Abstract
The Bruhat order is a well-studied partial order on Coxeter groups and Schubert varieties. Deodhar provided several characterizations of the Bruhat order, including the so-called “Z-property.” Another partial order on Coxeter groups that is relevant in combinatorics is the absolute order, which extends the notion of the classical noncrossing partitions to Coxeter groups. In this note, we prove a result that implies that the absolute order satisfies a weaker version of the Z-property.
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Acknowledgments
The author thanks Matthew J. Dyer for several useful discussions, as well as for his hospitality during visits to the University of Notre Dame, and for reading an earlier version of this note and providing suggestions on how to improve it. Furthermore, the author thanks an anonymous referee for invaluable comments on how to improve the presentation of the paper and for pointing out the current form of Theorem 10. Of course, all errors that remain are the author’s.
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Blanco, S.A. Weak Z-Property of the Absolute Order on Groups Generated by Sets Closed Under Taking Inverses. Order 36, 391–397 (2019). https://doi.org/10.1007/s11083-018-9474-3
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DOI: https://doi.org/10.1007/s11083-018-9474-3