The Hilbert series of the superspace coinvariant ring

Brendon Rhoades ; Andrew Timothy Wilson

Forum of Mathematics, Pi, Tome 12 (2024)

Résumé

Let $\Omega _n$ be the ring of polynomial-valued holomorphic differential forms on complex n-space, referred to in physics as the superspace ring of rank n. The symmetric group ${\mathfrak {S}}_n$ acts diagonally on $\Omega _n$ by permuting commuting and anticommuting generators simultaneously. We let $SI_n \subseteq \Omega _n$ be the ideal generated by ${\mathfrak {S}}_n$-invariants with vanishing constant term and study the quotient $SR_n = \Omega _n / SI_n$ of superspace by this ideal. We calculate the doubly-graded Hilbert series of $SR_n$ and prove an ‘operator theorem’, which characterizes the harmonic space $SH_n \subseteq \Omega _n$ attached to $SR_n$ in terms of the Vandermonde determinant and certain differential operators. Our methods employ commutative algebra results that were used in the study of Hessenberg varieties. Our results prove conjectures of N. Bergeron, Colmenarejo, Li, Machacek, Sulzgruber, Swanson, Wallach and Zabrocki.

Détail

DOI : 10.1017/fmp.2024.14

Bibliographie

[1] Abe, H., Harada, M., Horiguchi, T. and Masuda, M.. ‘The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A’, Int. Math. Res. Not. IMRN 17 (2019), 5316–5388.Google Scholar | DOI

[2] Abe, T., Horiguchi, T., Masuda, M., Murai, S. and Sato, T.. ‘Hessenberg varieties and hyperplane arrangements’, J. Reine Angew. Math. 764 (2020), 241–286.Google Scholar | DOI

[3] Angarone, R., Commins, P., Karn, T., Murai, S. and Rhoades, B., ‘Superspace coinvariants and hyperplane arrangements’, Preprint, 2024, .Google Scholar | arXiv

[4] Artin, E., Galois Theory (Notre Dame Math Lectures) no. 2, second edn. (Notre Dame, University of Notre Dame, 1944).Google Scholar

[5] Bergeron, F., ‘Multivariate diagonal coinvariant spaces for complex reflection groups’, Adv. Math. 239 (2013), 97–108.Google Scholar | DOI

[6] Bergeron, F., ‘-modules of multivariate diagonal harmonics’, Preprint, 2020, .Google Scholar | arXiv

[7] Bergeron, F., ‘The bosonic-fermionic diagonal coinvariant modules conjecture’, Preprint, 2020, .Google Scholar | arXiv

[8] Borel, A., ‘Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compass’, Ann. of Math. 57 (1953), 115–207.Google Scholar | DOI

[9] Carlsson, E. and Oblomkov, A., ‘Affine Schubert calculus and double coinvariants’, Preprint, 2018, .Google Scholar | arXiv

[10] Chevalley, C., ‘Invariants of finite groups generated by reflections’, Amer. J. Math. 77(4) (1955), 778–782.Google Scholar | DOI

[11] D’Adderio, M., Iraci, A. and Vanden Wyngaerd, A.. ‘Theta operators, refined Delta conjectures, and coinvariants’, Adv. Math. 376 (2021), 107477.Google Scholar

[12] Garsia, A. M. and Haiman, M., ‘A remarkable q, t-Catalan sequence and q-Lagrange inversion’, J. Algebraic Combin. 5(3) (1996), 191–244.Google Scholar | DOI

[13] Gillespie, M. and Rhoades, B., ‘Higher Specht bases for generalizations of the coinvariant ring’, Ann. Comb. 25(1) (2021), 51–77.Google Scholar | DOI

[14] Haglund, J., Remmel, J. and Wilson, A. T., ‘The Delta Conjecture’, Trans. Amer. Math. Soc. 370 (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. Loth. Comb. 79 (2019), Article B79c.Google Scholar

[17] Haiman, M., ‘Conjectures on the quotient ring by diagonal invariants’, J. Algebraic Combin. 3 (1994), 17–76.Google Scholar | DOI

[18] 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

[19] Harada, M., Horiguchi, T., Murai, S., Precup, M. and Tymoczko, J., ‘A filtration on the cohomology rings of regular nilpotent Hessenberg varieties’, Math. Z. 298 (2021), 1345–1382.Google Scholar | DOI

[20] Iraci, A., Rhoades, B. and Romero, M.. ‘A proof of the fermionic Theta coinvariant conjecture’, Preprint, 2022, .Google Scholar | arXiv | DOI

[21] Kim, J., ‘A combinatorial model for the fermionic diagonal coinvariant ring’, to appear, Combin. Theory 2022, .Google Scholar | arXiv | DOI

[22] Kim, J. and Rhoades, B., ‘Lefschetz theory for exterior algebras and fermionic diagonal coinvariants’, Int. Math. Res. Not. IMRN 2022(4) (2022), 2906–2933.Google Scholar | DOI

[23] Kim, J. and Rhoades, B., ‘Set partitions, fermions, and skein relations’, Int. Math. Res. Not. IMRN (2022), rnac110, https://doi.org/10.1093/imrn/rnac110.Google Scholar

[24] Maeno, T. and Watanabe, J., ‘Lefschetz elements of Artinian Gorenstein algebras and Hessians of homogeneous polynomials’, Illinois J. Math. 53(2) (2009), 591–603.Google Scholar | DOI

[25] Maeno, T., Numata, Y. and Wachi, A., ‘Strong Lefschetz elements of the coinvariant rings of finite Coxeter groups’, Algebr. Represent. Th. 14(4) (2007), 625–638.Google Scholar | DOI

[26] Meyer, K., ‘Descent representations for generalized coinvariant algebras’, Algebraic Combin. 3(4) (2020), 805–830.Google Scholar | DOI

[27] Pawlowski, B. and Rhoades, B., ‘A flag variety for the Delta Conjecture’, Trans. Amer. Math. Soc. 372(11) (2019), 8195–8248.Google Scholar | DOI

[28] Peskin, M. and Schroeder, D., An Introduction to Quantum Field Theory (Westview Press, 1995).Google Scholar

[29] Remmel, J. and Wilson, A. T., ‘An extension of MacMahon’s Equidistribution Theorem to ordered set partitions’, J. Combin. Theory Ser. A 34 (2015), 242–277.Google Scholar | DOI

[30] Rhoades, B. and Wilson, A. T., ‘Vandermondes in superspace’, Trans. Amer. Math. Soc. 373(6) (2020), 4483–4516.Google Scholar | DOI

[31] Rhoades, B. and Wilson, A. T., ‘Set superpartitions and superspace duality modules’, Accepted, Forum Math. Sigma 2022, .Google Scholar | arXiv | DOI

[32] Rhoades, B., Yu, T. and Zhao, Z., ‘Harmonic bases for generalized coinvariant algebras’, Electron. J. Combin. 27(4) (2020), P4.16.Google Scholar | DOI

[33] Sagan, B. and Swanson, J., ‘-Stirling numbers in type ’, Preprint, 2022, .Google Scholar | arXiv

[34] Solomon, L., ‘Invariants of finite reflection groups’, Nagoya Math. J. 22 (1963), 57–64.Google Scholar | DOI

[35] Swanson, J., ‘Tanisaki witness relations for harmonic differential forms’, Preprint, 2021, .Google Scholar | arXiv

[36] Swanson, J. and Wallach, N., ‘Harmonic differential forms for pseudo-reflection groups I. Semi-invariants’, J. Combin. Theory Ser. A. 182 (2021), no. 105474.Google Scholar | DOI

[37] Swanson, J. and Wallach, N., ‘Harmonic differential forms for pseudo-reflection groups II. Bi-degree bounds’, Preprint, 2021, .Google Scholar | arXiv | DOI

[38] Wallach, N., ‘Some implications of a conjecture of Zabrocki to the action of on polynomial differential forms’, Preprint, 2019, .Google Scholar | arXiv

[39] Wilson, A. T., ‘An extension of MacMahon’s Equidistribution Theorem to ordered multiset parititons’, Electron. J. Combin. 24(3) (2017), P3.21.Google Scholar

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