The Delta Conjecture at $q = 1$
Author:
Marino Romero
Journal:
Trans. Amer. Math. Soc. 369 (2017), 7509-7530
MSC (2010):
Primary 05E05, 05E10, 05Exx
DOI:
https://doi.org/10.1090/tran/7140
Published electronically:
June 27, 2017
MathSciNet review:
3683116
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We use a weight-preserving, sign-reversing involution to find a combinatorial expansion of $\Delta _{e_k} e_n$ at $q=1$ in terms of the elementary symmetric function basis. We then use a weight-preserving bijection to prove the Delta Conjecture of Haglund, Remmel, and Wilson at $q=1$. The method of proof provides a variety of structures which can compute the inner product of $\Delta _{e_k} e_n|_{q=1}$ with any symmetric function.
- A. M. Garsia and M. Haiman, A remarkable $q,t$-Catalan sequence and $q$-Lagrange inversion, J. Algebraic Combin. 5 (1996), no. 3, 191–244. MR 1394305, DOI https://doi.org/10.1023/A%3A1022476211638
- A. M. Garsia and M. Haiman, Some natural bigraded $S_n$-modules and $q,t$-Kostka coefficients, Electron. J. Combin. 3 (1996), no. 2, Research Paper 24, approx. 60. The Foata Festschrift. MR 1392509
- A. M. Garsia, G. Xin, and M. Zabrocki, Hall-Littlewood operators in the theory of parking functions and diagonal harmonics, Int. Math. Res. Not. IMRN 6 (2012), 1264–1299. MR 2899952, DOI https://doi.org/10.1093/imrn/rnr060
- A. Mellit, Toric braids and $(m,n)$-parking functions, arXiv:1604.07456 (2016).
- Andrew Timothy Wilson, A weighted sum over generalized Tesler matrices, J. Algebraic Combin. 45 (2017), no. 3, 825–855. MR 3627505, DOI https://doi.org/10.1007/s10801-016-0726-2
- Igor Burban and Olivier Schiffmann, On the Hall algebra of an elliptic curve, I, Duke Math. J. 161 (2012), no. 7, 1171–1231. MR 2922373, DOI https://doi.org/10.1215/00127094-1593263
- E. Carlsson and A. Mellit, A proof of the shuffle conjecture, arXiv:1508.06239 (2015).
- Eugene Gorsky and Andrei Neguţ, Refined knot invariants and Hilbert schemes, J. Math. Pures Appl. (9) 104 (2015), no. 3, 403–435 (English, with English and French summaries). MR 3383172, DOI https://doi.org/10.1016/j.matpur.2015.03.003
- Francois Bergeron, Adriano Garsia, Emily Sergel Leven, and Guoce Xin, Some remarkable new plethystic operators in the theory of Macdonald polynomials, J. Comb. 7 (2016), no. 4, 671–714. MR 3538159, DOI https://doi.org/10.4310/JOC.2016.v7.n4.a6
- F. Bergeron, A. M. Garsia, M. Haiman, and G. Tesler, Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions, Methods Appl. Anal. 6 (1999), no. 3, 363–420. Dedicated to Richard A. Askey on the occasion of his 65th birthday, Part III. MR 1803316, DOI https://doi.org/10.4310/MAA.1999.v6.n3.a7
- F. Bergeron and A. M. Garsia, Science fiction and Macdonald’s polynomials, Algebraic methods and $q$-special functions (Montréal, QC, 1996) CRM Proc. Lecture Notes, vol. 22, Amer. Math. Soc., Providence, RI, 1999, pp. 1–52. MR 1726826, DOI https://doi.org/10.1090/crmp/022/01
- J. Haglund, A proof of the $q,t$-Schröder conjecture, Int. Math. Res. Not. 11 (2004), 525–560. MR 2038776, DOI https://doi.org/10.1155/S1073792804132509
- I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
- Jim Haglund, The combinatorics of knot invariants arising from the study of Macdonald polynomials, Recent trends in combinatorics, IMA Vol. Math. Appl., vol. 159, Springer, [Cham], 2016, pp. 579–600. MR 3526424, DOI https://doi.org/10.1007/978-3-319-24298-9_23
- J. Haglund, J. B. Remmel, and A. T. Wilson, The Delta Conjecture, arXiv:1509.07058 (2015).
- J. Haglund, J. Morse, and M. Zabrocki, A compositional shuffle conjecture specifying touch points of the Dyck path, Canad. J. Math. 64 (2012), no. 4, 822–844. MR 2957232, DOI https://doi.org/10.4153/CJM-2011-078-4
- J. Haglund, M. Haiman, N. Loehr, J. B. Remmel, and A. Ulyanov, A combinatorial formula for the character of the diagonal coinvariants, Duke Math. J. 126 (2005), no. 2, 195–232. MR 2115257, DOI https://doi.org/10.1215/S0012-7094-04-12621-1
- Mark Haiman, Hilbert schemes, polygraphs and the Macdonald positivity conjecture, J. Amer. Math. Soc. 14 (2001), no. 4, 941–1006. MR 1839919, DOI https://doi.org/10.1090/S0894-0347-01-00373-3
- Mark Haiman, Combinatorics, symmetric functions, and Hilbert schemes, Current developments in mathematics, 2002, Int. Press, Somerville, MA, 2003, pp. 39–111. MR 2051783
- Ömer Eğecioğlu and Jeffrey B. Remmel, Brick tabloids and the connection matrices between bases of symmetric functions, Discrete Appl. Math. 34 (1991), no. 1-3, 107–120. Combinatorics and theoretical computer science (Washington, DC, 1989). MR 1137989, DOI https://doi.org/10.1016/0166-218X%2891%2990081-7
- Tewodros Amdeberhan and Emily Sergel Leven, Multi-cores, posets, and lattice paths, Adv. in Appl. Math. 71 (2015), 1–13. MR 3406955, DOI https://doi.org/10.1016/j.aam.2015.08.002
Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 05E05, 05E10, 05Exx
Retrieve articles in all journals with MSC (2010): 05E05, 05E10, 05Exx
Additional Information
Marino Romero
Affiliation:
Department of Mathematics, University of California, San Diego, La Jolla, California 92093
Email:
mar007@ucsd.edu
Received by editor(s):
September 14, 2016
Received by editor(s) in revised form:
November 23, 2016
Published electronically:
June 27, 2017
Additional Notes:
This research was supported by NSF grant 1362160
Article copyright:
© Copyright 2017
American Mathematical Society
