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A combinatorial $ \mathfrak{sl}_2$-action and the Sperner property for the weak order


Authors: Christian Gaetz and Yibo Gao
Journal: Proc. Amer. Math. Soc. 148 (2020), 1-7
MSC (2010): Primary 06A07, 06A11, 05E18
DOI: https://doi.org/10.1090/proc/14655
Published electronically: July 30, 2019
MathSciNet review: 4042823
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Abstract: We construct a simple combinatorially-defined representation of $ \mathfrak{sl}_2$ which respects the order structure of the weak order on the symmetric group. This is used to prove that the weak order has the strong Sperner property, and is therefore a Peck poset, solving a problem raised by Björner [Orderings of Coxeter groups, Amer. Math. Soc., Providence, RI, 1984, pp. 175-195]; a positive answer to this question had been conjectured by Stanley [Some Schubert shenanigans, preprint, 2017].


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Additional Information

Christian Gaetz
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: gaetz@mit.edu

Yibo Gao
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: gaoyibo@mit.edu

DOI: https://doi.org/10.1090/proc/14655
Keywords: Weak Bruhat order, Sperner, Peck
Received by editor(s): December 4, 2018
Received by editor(s) in revised form: April 3, 2019
Published electronically: July 30, 2019
Additional Notes: The first author was partially supported by an NSF Graduate Research Fellowship.
An extended abstract of this work will appear in the proceedings of FPSAC 2019
Communicated by: Patricia L. Hersh
Article copyright: © Copyright 2019 American Mathematical Society