A Proof of the Extended Delta Conjecture

Jonah Blasiak ; Mark Haiman ; Jennifer Morse ; Anna Pun ; George H. Seelinger

Forum of Mathematics, Pi, Tome 11 (2023)

Résumé

We prove the extended delta conjecture of Haglund, Remmel and Wilson, a combinatorial formula for $\Delta _{h_l}\Delta ' _{e_k} e_{n}$, where $\Delta ' _{e_k}$ and $\Delta _{h_l}$ are Macdonald eigenoperators and $e_n$ is an elementary symmetric function. We actually prove a stronger identity of infinite series of $\operatorname {\mathrm {GL}}_m$ characters expressed in terms of LLT series. This is achieved through new results in the theory of the Schiffmann algebra and its action on the algebra of symmetric functions.

Détail

DOI : 10.1017/fmp.2023.3

Bibliographie

[1] Bergeron, F., Garsia, A. M., Haiman, M., and Tesler, G., ‘Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions’, Methods Appl. Anal. 6(3) (1999), 363–420. Dedicated to Richard A. Askey on the occasion of his 65th birthday, Part III.Google Scholar | DOI

[2] Blasiak, J., Haiman, M., Morse, J., Pun, A. and Seelinger, G. H., ‘A shuffle theorem for paths under any line’, Forum of Mathematics, Pi. 11, E5.Google Scholar | DOI

[3] Burban, I. and Schiffmann, O., ‘On the Hall algebra of an elliptic curve, I’, Duke Math. J. 161(7) (2012), 1171–1231.Google Scholar | DOI

[4] Carlsson, E. and Mellit, A., ‘A proof of the shuffle conjecture’, J. Amer. Math. Soc. 31(3) (2018), 661–697.Google Scholar | DOI

[5] D’Adderio, M., Iraci, A. and Vanden Wyngaerd, A., ‘The Schröder case of the generalized Delta conjecture’, European J. Combin. 81 (2019), 58–83.Google Scholar | DOI

[6] D’Adderio, M., Iraci, A. and Vanden Wyngaerd, A., ‘Theta operators, refined delta conjectures, and coinvariants’, Adv. Math. 376 (2021), 107447.Google Scholar | DOI

[7] D’Adderio, M. and Mellit, A., ‘A proof of the compositional Delta conjecture’, Adv. Math. 402 (2022), 108342.Google Scholar | DOI

[8] Feigin, B. L. and Tsymbaliuk, A. I., ‘Equivariant -theory of Hilbert schemes via shuffle algebra’, Kyoto J. Math. 51(4) (2011), 831–854.Google Scholar

[9] Garsia, A. M. and Haiman, M., ‘Some natural bigraded -modules and -Kostka coefficients’, Electron. J. Combin. 3(2) (1996), Research Paper 24, The Foata Festschrift.Google Scholar | DOI

[10] Garsia, A. M., Haiman, M. and Tesler, G., ‘Explicit plethystic formulas for Macdonald -Kostka coefficients’, Sém. Lothar. Combin. 42 (1999), Art. B42m, The Andrews Festschrift (Maratea, 1998).Google Scholar

[11] Garsia, A., Haglund, J., Remmel, J. B. and Yoo, Meesue, ‘A proof of the delta conjecture when ’, Ann. Comb. 23(2) (2019), 317–333.Google Scholar | DOI

[12] Grojnowski, I. and Haiman, M., ‘ Affine Hecke algebras and positivity of LLT and Macdonald polynomials ’, Unpublished manuscript, 2007.Google Scholar

[13] Haglund, J., Haiman, M., Loehr, N., Remmel, J. B. and Ulyanov, A., ‘A combinatorial formula for the character of the diagonal coinvariants’, Duke Math. J. 126(2) (2005), 195–232.Google Scholar | DOI

[14] Haglund, J., Remmel, J. B. and Wilson, A. T., ‘The delta conjecture’, Trans. Amer. Math. Soc. 370(6) (2018), 4029–4057.Google Scholar | DOI

[15] Haglund, J., Rhoades, B. and Shimozono, M., ‘Ordered set partitions, generalized coinvariant algebras, and the delta conjecture’, Adv. Math. 329 (2018), 851–915.Google Scholar | DOI

[16] Haglund, J., Rhoades, B. and Shimozono, M., ‘Hall–Littlewood expansions of Schur delta operators at ’, Sém. Lothar. Combin. 79 (2018–2019), Art. B79c.Google Scholar

[17] Haiman, M., ‘Vanishing theorems and character formulas for the Hilbert scheme of points in the plane’, Invent. Math. 149(2) (2002), 371–407.Google Scholar | DOI

[18] Macdonald, I. G., ‘Symmetric Functions and Hall Polynomials’, second edn. (The Clarendon Press, Oxford University Press, New York, 1995), With contributions by A. Zelevinsky, Oxford Science Publications.Google Scholar

[19] Negut, A., ‘The shuffle algebra revisited’, Int. Math. Res. Not. IMRN (22) (2014), 6242–6275.Google Scholar | DOI

[20] Negut, A., ‘Hecke correspondences for smooth moduli spaces of sheaves’, Publ. Math. Inst. Hautes Études Sci. 135 (2022), 337–418.Google Scholar | DOI

[21] Qiu, D. and Wilson, A. T., ‘The valley version of the extended delta conjecture’, J. Combin. Theory Ser. A 175 (2020), 105271.Google Scholar | DOI

[22] Rhoades, B., ‘Ordered set partition statistics and the delta conjecture’, J. Combin. Theory Ser. A 154 (2018), 172–217.Google Scholar | DOI

[23] Romero, M., ‘The delta conjecture at ’, Trans. Amer. Math. Soc. 369(10) (2017), 7509–7530.Google Scholar | DOI

[24] Schiffmann, O., ‘On the Hall algebra of an elliptic curve, II’, Duke Math. J. 161(9) (2012), 1711–1750.Google Scholar | DOI

[25] Schiffmann, O. and Vasserot, E., ‘The elliptic Hall algebra and the -theory of the Hilbert scheme of ’, Duke Math. J. 162(2) (2013), 279–366.Google Scholar | DOI

[26] Wilson, A. T., ‘A weighted sum over generalized Tesler matrices’, J. Algebraic Combin. 45(3) (2017), 825–855.Google Scholar | DOI

[27] Zabrocki, M., ‘A module for the delta conjecture’, Preprint, 2019, .Google Scholar | arXiv

[28] Zabrocki, M., ‘A proof of the 4-variable Catalan polynomial of the delta conjecture’, J. Comb. 10(4) (2019), 599–632.Google Scholar