ALGEBRAIC COMBINATORICS

no. 1
Tanisaki witness relations for harmonic differential forms
Swanson, Joshua P.1
1 University of Southern California Department of mathematics Los Angeles CA 90089 (USA)
Algebraic Combinatorics, Volume 7 (2024) no. 1, pp. 159-185.

Inspired by a series of conjectures and formulas related to higher coinvariant algebras, we present two families of relations involving harmonic differential forms of the symmetric group. Our relations, together with a novel bijection, are sufficient to give a filtration of the 1-forms suggested by work of Haglund–Rhoades–Shimozono with composition factors given by Tanisaki quotients. These are “almost all” of the necessary relations in a certain asymptotic sense we make precise.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.334
Classification: 05E10, 05E16
Keywords: coinvariant algebras, Delta Conjecture, Tanisaki ideals, differential harmonics
Swanson, Joshua P. 1

1 University of Southern California Department of mathematics Los Angeles CA 90089 (USA)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Swanson, Joshua P. Tanisaki witness relations  for harmonic differential forms. Algebraic Combinatorics, Volume 7 (2024) no. 1, pp. 159-185. doi : 10.5802/alco.334. https://alco.centre-mersenne.org/articles/10.5802/alco.334/

[1] Alfano, Joseph A basis for the Y 1 subspace of diagonal harmonic polynomials, Discrete Math., Volume 193 (1998) no. 1-3, pp. 17-31 | DOI | MR | Zbl

[2] Bergeron, François Algebraic combinatorics and coinvariant spaces, CMS Treatises in Mathematics, Canadian Mathematical Society, Ottawa, ON; A K Peters, Ltd., Wellesley, MA, 2009, viii+221 pages | DOI | MR

[3] Blasiak, Jonah; Haiman, Mark; Morse, Jennifer; Pun, Anna; Seelinger, George H. A proof of the Extended Delta Conjecture, Forum Math. Pi, Volume 11 (2023), Paper no. e6, 28 pages | DOI | MR | Zbl

[4] Borel, Armand Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math. (2), Volume 57 (1953), pp. 115-207 | DOI | MR | Zbl

[5] D’Adderio, Michele; Mellit, Anton A proof of the compositional Delta conjecture, Adv. Math., Volume 402 (2022), Paper no. 108342, 17 pages | DOI | MR | Zbl

[6] Fulton, William Young tableaux: with applications to representation theory and geometry, London Mathematical Society Student Texts, 35, Cambridge University Press, Cambridge, 1997, x+260 pages | MR

[7] Garsia, A. M.; Procesi, C. On certain graded S n -modules and the q-Kostka polynomials, Adv. Math., Volume 94 (1992) no. 1, pp. 82-138 | DOI | MR | Zbl

[8] Garsia, Adriano M. Combinatorial methods in the theory of Cohen-Macaulay rings, Adv. in Math., Volume 38 (1980) no. 3, pp. 229-266 | DOI | MR | Zbl

[9] Haglund, J.; Remmel, J. B.; Wilson, A. T. The Delta Conjecture, Trans. Amer. Math. Soc., Volume 370 (2018) no. 6, pp. 4029-4057 | DOI | MR | Zbl

[10] Haglund, James; Rhoades, Brendon; Shimozono, Mark Ordered set partitions, generalized coinvariant algebras, and the Delta Conjecture, Adv. Math., Volume 329 (2018), pp. 851-915 | DOI | MR | Zbl

[11] 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

[12] Haiman, Mark D. Conjectures on the quotient ring by diagonal invariants, J. Algebraic Combin., Volume 3 (1994) no. 1, pp. 17-76 | DOI | MR | Zbl

[13] Humphreys, James E. Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972, xii+169 pages | DOI | MR

[14] Pawlowski, Brendan; Rhoades, Brendon A flag variety for the Delta Conjecture, Trans. Amer. Math. Soc., Volume 372 (2019) no. 11, pp. 8195-8248 | DOI | MR | Zbl

[15] Rhoades, Brendon; Wilson, Andrew Timothy Vandermondes in superspace, Trans. Amer. Math. Soc., Volume 373 (2020) no. 6, pp. 4483-4516 | DOI | MR | Zbl

[16] Rhoades, Brendon; Wilson, Andrew Timothy The Hilbert series of the superspace coinvariant ring, 2023 | arXiv

[17] Solomon, Louis Invariants of finite reflection groups, Nagoya Math. J., Volume 22 (1963), pp. 57-64 http://projecteuclid.org/euclid.nmj/1118801157 | DOI | MR | Zbl

[18] Stanley, Richard P. Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc. (N.S.), Volume 1 (1979) no. 3, pp. 475-511 | DOI | MR | Zbl

[19] Steinberg, Robert Differential equations invariant under finite reflection groups, Trans. Amer. Math. Soc., Volume 112 (1964), pp. 392-400 | DOI | MR | Zbl

[20] Swanson, Joshua P.; Wallach, Nolan R. Harmonic differential forms for pseudo-reflection groups I. Semi-invariants, J. Combin. Theory Ser. A, Volume 182 (2021), Paper no. 105474, 30 pages | DOI | MR | Zbl

[21] Swanson, Joshua P.; Wallach, Nolan R. Harmonic differential forms for pseudo-reflection groups II. Bi-degree bounds, 2023 | arXiv

[22] Tanisaki, Toshiyuki Defining ideals of the closures of the conjugacy classes and representations of the Weyl groups, Tohoku Math. J. (2), Volume 34 (1982) no. 4, pp. 575-585 | DOI | MR | Zbl

[23] Zabrocki, Mike A module for the Delta conjecture, 2019 | arXiv

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