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no. 6
Stable-limit non-symmetric Macdonald functions
Bechtloff Weising, Milo J.1
1 University of California, Davis Dept. of Mathematics One Shields Avenue Davis CA 95616
Algebraic Combinatorics, Volume 7 (2024) no. 6, pp. 1845-1878.

We construct and study an explicit simultaneous 𝒴-eigenbasis of Ion and Wu’s standard representation of the + stable-limit double affine Hecke algebra for the limit Cherednik operators 𝒴 i . This basis arises as a variant of Cherednik’s non-symmetric Macdonald polynomials of type GL. We utilize links between + stable-limit double affine Hecke algebra theory of Ion–Wu and the double Dyck path algebra of Carlsson–Mellit that arose in their proof of the Shuffle Conjecture. As a consequence, the spectral theory for the limit Cherednik operators is understood. The symmetric functions comprise the zero weight space. We introduce one extra operator that commutes with the 𝒴 i action and dramatically refines the weight spaces to now be one-dimensional. This operator, up to a change of variables, gives an extension of Haiman’s operator Δ ′ from Λ to 𝒫 as + . Additionally, we develop another method to build this weight basis using limits of trivial idempotents.

Received:
Accepted:
Published online:
DOI: 10.5802/alco.395
Classification: 05E05, 33D52
Mots-clés : Macdonald polynomials, double affine Hecke algebras, stable-limit

Bechtloff Weising, Milo J. 1

1 University of California, Davis Dept. of Mathematics One Shields Avenue Davis CA 95616
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Bechtloff Weising, Milo J. Stable-limit non-symmetric Macdonald functions. Algebraic Combinatorics, Volume 7 (2024) no. 6, pp. 1845-1878. doi : 10.5802/alco.395. https://alco.centre-mersenne.org/articles/10.5802/alco.395/

[1] Bechtloff Weising, Milo James Stable-limit non-symmetric Macdonald functions in type A, Sém. Lothar. Combin., Volume 89B (2023), Paper no. 56, 12 pages | MR

[2] Bergeron, F.; Garsia, A. M. Science fiction and Macdonald’s polynomials, Algebraic methods and q-special functions (Montréal, QC, 1996) (CRM Proc. Lecture Notes), Volume 22, Amer. Math. Soc., Providence, RI, 1999, pp. 1-52 | DOI | MR | Zbl

[3] Bergeron, F.; Garsia, A. M.; Haiman, M.; Tesler, G. Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions, Methods Appl. Anal., Volume 6 (1999) no. 3, pp. 363-420 | DOI | MR | Zbl

[4] Burban, Igor; Schiffmann, Olivier On the Hall algebra of an elliptic curve, I, Duke Math. J., Volume 161 (2012) no. 7, pp. 1171-1231 | DOI | MR | Zbl

[5] Carlsson, Erik; Gorsky, Eugene; Mellit, Anton The 𝔸 q,t algebra and parabolic flag Hilbert schemes, Math. Ann., Volume 376 (2020) no. 3-4, pp. 1303-1336 | DOI | MR | Zbl

[6] Carlsson, Erik; Mellit, Anton A proof of the shuffle conjecture, J. Amer. Math. Soc., Volume 31 (2018) no. 3, pp. 661-697 | DOI | MR | Zbl

[7] Cherednik, Ivan Double affine Hecke algebras and difference Fourier transforms, Invent. Math., Volume 152 (2003) no. 2, pp. 213-303 | DOI | MR | Zbl

[8] Garsia, A. M.; Haiman, M. A remarkable q,t-Catalan sequence and q-Lagrange inversion, J. Algebraic Combin., Volume 5 (1996) no. 3, pp. 191-244 | DOI | MR | Zbl

[9] Goodberry, Ben Partially-Symmetric Macdonald Polynomials, Ph. D. Thesis, Virginia Polytechnic Institute and State University (2022)

[10] Haglund, J.; Haiman, M.; Loehr, N. A combinatorial formula for nonsymmetric Macdonald polynomials, Amer. J. Math., Volume 130 (2008) no. 2, pp. 359-383 | DOI | MR | Zbl

[11] Haglund, J.; Haiman, M.; Loehr, N.; Remmel, J. B.; Ulyanov, A. A combinatorial formula for the character of the diagonal coinvariants, Duke Math. J., Volume 126 (2005) no. 2, pp. 195-232 | DOI | MR | Zbl

[12] Haglund, J.; Morse, J.; Zabrocki, M. A compositional shuffle conjecture specifying touch points of the Dyck path, Canad. J. Math., Volume 64 (2012) no. 4, pp. 822-844 | DOI | MR | Zbl

[13] Haiman, Mark Macdonald polynomials and geometry, New perspectives in algebraic combinatorics (Berkeley, CA, 1996–97) (Math. Sci. Res. Inst. Publ.), Volume 38, Cambridge Univ. Press, Cambridge, 1999, pp. 207-254 | MR | Zbl

[14] Haiman, Mark Vanishing theorems and character formulas for the Hilbert scheme of points in the plane, Invent. Math., Volume 149 (2002) no. 2, pp. 371-407 | DOI | MR | Zbl

[15] Ion, Bogdan; Wu, Dongyu Unpublished notes, 2022

[16] Ion, Bogdan; Wu, Dongyu The stable limit DAHA and the double Dyck path algebra, J. Inst. Math. Jussieu, Volume 23 (2024) no. 1, pp. 379-424 | DOI | MR | Zbl

[17] Jing, Nai Huan Vertex operators and Hall-Littlewood symmetric functions, Adv. Math., Volume 87 (1991) no. 2, pp. 226-248 | DOI | MR | Zbl

[18] Lapointe, Luc m-Symmetric functions, non-symmetric Macdonald polynomials and positivity conjectures, 2022 | arXiv

[19] Macdonald, I. G. Symmetric functions and Hall polynomials, Oxford Classic Texts in the Physical Sciences, The Clarendon Press, Oxford University Press, New York, 2015, xii+475 pages | MR

[20] Ram, Arun Alcove walks, Hecke algebras, spherical functions, crystals and column strict tableaux, Pure Appl. Math. Q., Volume 2 (2006) no. 4, pp. 963-1013 | DOI | MR | Zbl

[21] Schiffmann, Olivier; Vasserot, Eric The elliptic Hall algebra and the K-theory of the Hilbert scheme of 𝔸 2 , Duke Math. J., Volume 162 (2013) no. 2, pp. 279-366 | DOI | MR | Zbl

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