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The Newton polytope and Lorentzian property of chromatic symmetric functions

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Abstract

Chromatic symmetric functions are well-studied symmetric functions in algebraic combinatorics that generalize the chromatic polynomial and are related to Hessenberg varieties and diagonal harmonics. Motivated by the Stanley–Stembridge conjecture, we show that the allowable coloring weights for indifference graphs of Dyck paths are the lattice points of a permutahedron \(\mathcal {P}_\lambda \), and we give a formula for the dominant weight \(\lambda \). Furthermore, we conjecture that such chromatic symmetric functions are Lorentzian, a property introduced by Brändén and Huh as a bridge between discrete convex analysis and concavity properties in combinatorics, and we prove this conjecture for abelian Dyck paths. We extend our results on the Newton polytope to incomparability graphs of \((3+1)\)-free posets, and we give a number of conjectures and results stemming from our work, including results on the complexity of computing the coefficients and relations with the \(\zeta \) map from diagonal harmonics.

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Notes

  1. The strongest version of Mason’s conjecture was also proved independently and simultaneously in [7].

  2. In [30] the authors use the letter \(c_i\) corresponding to clones that correspond to consecutive copies of the letter \(v_i\) in the part listing.

  3. A previous version of the current paper incorrectly restated the definition of the canonical part listing from Guay-Paquet et al. [30].

  4. This conjecture has been proved independently by Gélinas–Segovia–Thomas [27] and by Fang [22].

  5. There are no 3-antichains of partitions of \(n<12\) in dominance order with one partition being a convex combination of the other two.

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Acknowledgements

This project started during a stay of the second named author at Institut Mittag-Leffler in Djursholm, Sweden in the course of the program on Algebraic and Enumerative Combinatorics in Spring 2020. We thankfully acknowledge the support of the Swedish Research Council under Grant No. 2016-06596, and thank Institut Mittag-Leffler for its hospitality. We thank Margaret Bayer, Petter Brändén, Tim Chow, Laura Colmenarejo, Benedek Dombos, Félix Gélinas, Mathieu Guay-Paquet, Álvaro Gutiérrez, Jim Haglund, Chris Hanusa, June Huh, Khanh Nguyen Duc, Greta Panova, Adrien Segovia, Mark Skandera, Eric Sommers, Hugh Thomas, and Andrew Tymothy Wilson for helpful comments. We also would like to thank the anonymous referee for edits and suggestions that improved the manuscript. The first named author was partially supported by Max Planck Institute for Mathematics (MPIM), the Hausdorff Research Institute for Mathematics (HIM), and the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy—GZ 2047/1, Projekt-ID 390685813. The second and third named authors were partially supported by NSF Grants DMS-1855536 and DMS-2154019.

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Authors and Affiliations

  1. Department of Mathematics, North Carolina State University, Raleigh, NC, USA

    Jacob P. Matherne

  2. LACIM, Département de Mathématiques, Université du Québec à Montréal, Montréal, QC, Canada

    Alejandro H. Morales

  3. Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, USA

    Alejandro H. Morales & Jesse Selover

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  1. Jacob P. Matherne
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Correspondence to Alejandro H. Morales.

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Matherne, J.P., Morales, A.H. & Selover, J. The Newton polytope and Lorentzian property of chromatic symmetric functions. Sel. Math. New Ser. 30, 42 (2024). https://doi.org/10.1007/s00029-024-00928-4

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