Equivariant Kazhdan-Lusztig polynomials of thagomizer matroids
HTML articles powered by AMS MathViewer
- by Matthew H. Y. Xie and Philip B. Zhang
- Proc. Amer. Math. Soc. 147 (2019), 4687-4695
- DOI: https://doi.org/10.1090/proc/14608
- Published electronically: June 10, 2019
- PDF | Request permission
Abstract:
The equivariant Kazhdan-Lusztig polynomial of a matroid was introduced by Gedeon, Proudfoot, and Young. Gedeon conjectured an explicit formula for the equivariant Kazhdan-Lusztig polynomials of thagomizer matroids with an action of symmetric groups. In this paper, we discover a new formula for these polynomials which is related to the equivariant Kazhdan-Lusztig polynomials of uniform matroids. Based on our new formula, we confirm Gedeon’s conjecture by the Pieri rule.References
- Ben Elias, Nicholas Proudfoot, and Max Wakefield, The Kazhdan-Lusztig polynomial of a matroid, Adv. Math. 299 (2016), 36–70. MR 3519463, DOI 10.1016/j.aim.2016.05.005
- Katie R. Gedeon, Kazhdan-Lusztig polynomials of thagomizer matroids, Electron. J. Combin. 24 (2017), no. 3, Paper No. 3.12, 10. MR 3691529, DOI 10.37236/6567
- Katie Gedeon, Nicholas Proudfoot, and Benjamin Young, The equivariant Kazhdan-Lusztig polynomial of a matroid, J. Combin. Theory Ser. A 150 (2017), 267–294. MR 3645577, DOI 10.1016/j.jcta.2017.03.007
- Katie Gedeon, Nicholas Proudfoot, and Benjamin Young, Kazhdan-Lusztig polynomials of matroids: a survey of results and conjectures, Sém. Lothar. Combin. 78B (2017), Art. 80, 12. MR 3678662
- James Haglund, The $q$,$t$-Catalan numbers and the space of diagonal harmonics, University Lecture Series, vol. 41, American Mathematical Society, Providence, RI, 2008. With an appendix on the combinatorics of Macdonald polynomials. MR 2371044, DOI 10.1007/s10711-008-9270-0
- Mark Haiman and Alexander Woo, Geometry of $q$ and $q,t$-analogs in combinatorial enumeration, Geometric combinatorics, IAS/Park City Math. Ser., vol. 13, Amer. Math. Soc., Providence, RI, 2007, pp. 207–248. MR 2383128, DOI 10.1090/pcms/013/05
- I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Classic Texts in the Physical Sciences, The Clarendon Press, Oxford University Press, New York, 2015. With contribution by A. V. Zelevinsky and a foreword by Richard Stanley; Reprint of the 2008 paperback edition [ MR1354144]. MR 3443860
- N. Proudfoot, Equivariant Kazhdan-Lusztig polynomials of $q$-niform matroids, arXiv:1808.07855, 2018.
- Nicholas Proudfoot, Max Wakefield, and Ben Young, Intersection cohomology of the symmetric reciprocal plane, J. Algebraic Combin. 43 (2016), no. 1, 129–138. MR 3439303, DOI 10.1007/s10801-015-0628-8
- Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282, DOI 10.1017/CBO9780511609589
Bibliographic Information
- Matthew H. Y. Xie
- Affiliation: College of Science, Tianjin University of Technology, Tianjin 300384, People’s Republic of China
- MR Author ID: 1066556
- Email: xie@mail.nankai.edu.cn
- Philip B. Zhang
- Affiliation: College of Mathematical Science, Tianjin Normal University, Tianjin 300387, People’s Republic of China
- MR Author ID: 1066440
- Email: zhang@tjnu.edu.cn
- Received by editor(s): February 4, 2019
- Received by editor(s) in revised form: February 26, 2019
- Published electronically: June 10, 2019
- Additional Notes: This work was supported by the National Science Foundation of China (Nos. 11701424, 11801447).
The second author is the corresponding author - Communicated by: Yuan Xu
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 4687-4695
- MSC (2010): Primary 05B35, 05E05, 20C30
- DOI: https://doi.org/10.1090/proc/14608
- MathSciNet review: 4011505