A combinatorial $\mathfrak {sl}_2$-action and the Sperner property for the weak order
HTML articles powered by AMS MathViewer
- by Christian Gaetz and Yibo Gao PDF
- Proc. Amer. Math. Soc. 148 (2020), 1-7 Request permission
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].References
- Anders Björner, Orderings of Coxeter groups, Combinatorics and algebra (Boulder, Colo., 1983) Contemp. Math., vol. 34, Amer. Math. Soc., Providence, RI, 1984, pp. 175–195. MR 777701, DOI 10.1090/conm/034/777701
- Christian Gaetz and Yibo Gao, A combinatorial duality between the weak and strong Bruhat orders, arXiv:1812.05126 [math.CO], 2018.
- Christian Gaetz and Yibo Gao, The weak Bruhat order on the symmetric group is Sperner, Séminaire Lotharingien de Combinatoire, in press, 2019.
- Zachary Hamaker, Oliver Pechenik, David E Speyer, and Anna Weigandt, Derivatives of Schubert polynomials and proof of a determinant conjecture of Stanley, arXiv:1812.00321 [math.CO], 2018.
- Alexander Kirillov Jr., An introduction to Lie groups and Lie algebras, Cambridge Studies in Advanced Mathematics, vol. 113, Cambridge University Press, Cambridge, 2008. MR 2440737, DOI 10.1017/CBO9780511755156
- Robert A. Proctor, Representations of ${\mathfrak {s}}{\mathfrak {l}}(2,\,\textbf {C})$ on posets and the Sperner property, SIAM J. Algebraic Discrete Methods 3 (1982), no. 2, 275–280. MR 655567, DOI 10.1137/0603026
- Robert A. Proctor, Michael E. Saks, and Dean G. Sturtevant, Product partial orders with the Sperner property, Discrete Math. 30 (1980), no. 2, 173–180. MR 566433, DOI 10.1016/0012-365X(80)90118-1
- R. P. Stanley, Some Schubert shenanigans, arXiv:1704.00851 [math.CO], 2017.
- Richard P. Stanley, Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Algebraic Discrete Methods 1 (1980), no. 2, 168–184. MR 578321, DOI 10.1137/0601021
- Richard P. Stanley, Enumerative combinatorics. Volume 1, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 2012. MR 2868112
Additional Information
- Christian Gaetz
- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
- MR Author ID: 1156664
- ORCID: 0000-0002-3748-4008
- Email: gaetz@mit.edu
- Yibo Gao
- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
- MR Author ID: 1283652
- Email: gaoyibo@mit.edu
- 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
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 1-7
- MSC (2010): Primary 06A07, 06A11, 05E18
- DOI: https://doi.org/10.1090/proc/14655
- MathSciNet review: 4042823