ALGEBRAIC COMBINATORICS

On the Todd class of the permutohedral variety
Algebraic Combinatorics, Volume 4 (2021) no. 3, pp. 387-407.

In the special case of braid fans, we give a combinatorial formula for the Berline–Vergne’s construction for an Euler–Maclaurin type formula that computes the number of lattice points in polytopes. Our formula is obtained by computing a symmetric expression for the Todd class of the permutohedral variety. By showing that this formula does not always have positive values, we prove that the Todd class of the permutohedral variety ${X}_{d}$ is not effective for $d\ge 24$.

Additionally, we prove that the linear coefficient in the Ehrhart polynomial of any lattice generalized permutohedron is positive.

Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/alco.157
Classification: 52B20,  14M25
Keywords: Ehrhart polynomials, generalized permutohedra, Berline–Vergne construction
@article{ALCO_2021__4_3_387_0,
author = {Castillo, Federico and Liu, Fu},
title = {On the {Todd} class of the permutohedral variety},
journal = {Algebraic Combinatorics},
pages = {387--407},
publisher = {MathOA foundation},
volume = {4},
number = {3},
year = {2021},
doi = {10.5802/alco.157},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.157/}
}
Castillo, Federico; Liu, Fu. On the Todd class of the permutohedral variety. Algebraic Combinatorics, Volume 4 (2021) no. 3, pp. 387-407. doi : 10.5802/alco.157. https://alco.centre-mersenne.org/articles/10.5802/alco.157/

[1] Berline, Nicole; Vergne, Michèle The equivariant Todd genus of a complete toric variety, with Danilov condition, J. Algebra, Volume 313 (2007) no. 1, pp. 28-39 | Article | MR 2326136 | Zbl 1204.52016

[2] Berline, Nicole; Vergne, Michèle Local Euler–Maclaurin formula for polytopes, Mosc. Math. J., Volume 7 (2007) no. 3, p. 355-386, 573 | Article | MR 2343137 | Zbl 1014.52001

[3] Castillo, Federico; Liu, Fu Berline–Vergne valuation and generalized permutohedra, Discrete Comput. Geom., Volume 60 (2018) no. 4, pp. 885-908 | Article | MR 3869454 | Zbl 1154.52009

[4] Castillo, Federico; Liu, Fu Deformation Cones of nested Braid fans (2018) (https://arxiv.org/abs/1710.01899) | MR 1731815 | Zbl 0940.05004

[5] Castillo, Federico; Liu, Fu; Nill, Benjamin; Paffenholz, Andreas Smooth polytopes with negative Ehrhart coefficients, J. Combin. Theory Ser. A, Volume 160 (2018), pp. 316-331 | Article | MR 3846206 | Zbl 1126.14058

[6] Danilov, Vladimir I. Geometry of toric varieties, Russ. Math. Surv., Volume 33 (1978) no. 2, pp. 97-154 | Article | MR 2343137 | Zbl 0425.14013

[7] De Loera, Jesús A.; Haws, David C.; Köppe, Matthias Ehrhart polynomials of matroid polytopes and polymatroids, Discrete Comput. Geom., Volume 42 (2009) no. 4, pp. 670-702 | Article | MR 2556462

[8] Ehrhart, Eugène Sur les polyèdres rationnels homothétiques à $n$ dimensions, C. R. Acad. Sci. Paris, Volume 254 (1962), pp. 616-618 | MR 0130860 | Zbl 1401.52024

[9] Eisenbud, David; Harris, Joe The geometry of schemes, Graduate Texts in Mathematics, 197, Springer-Verlag, New York, 2000, x+294 pages | Article | MR 1730819 | Zbl 1401.52024

[10] Ewald, Günter Combinatorial convexity and algebraic geometry, Graduate Texts in Mathematics, 168, Springer-Verlag, New York, 1996, xiv+372 pages | Article | MR 1418400

[11] Ferroni, Luis Hypersimplices are Ehrhart positive, J. Combin. Theory Ser. A, Volume 178 (2021), Paper no. 105365, 13 pages | Article | MR 4179055 | Zbl 1402.52017

[12] Fulton, William Introduction to toric varieties, Annals of Mathematics Studies, 131, Princeton University Press, Princeton, NJ, 1993, xii+157 pages (The William H. Roever Lectures in Geometry) | Article | MR 1234037 | Zbl 0425.14013

[13] Garoufalidis, Stavros; Pommersheim, James Sum-integral interpolators and the Euler–Maclaurin formula for polytopes, Trans. Amer. Math. Soc., Volume 364 (2012) no. 6, pp. 2933-2958 | Article | MR 2888234 | Zbl 1207.52015

[14] Hibi, Takayuki; Higashitani, Akihiro; Tsuchiya, Akiyoshi; Yoshida, Koutarou Ehrhart polynomials with negative coefficients, Graphs Combin., Volume 35 (2019) no. 1, pp. 363-371 | Article | MR 3898396 | Zbl 0100.27601

[15] Jochemko, Katharina; Ravichandran, Mohan Generalized permutahedra: Minkowski linear functionals and Ehrhart positivity (2019) (https://arxiv.org/abs/1909.08448) | MR 1730819 | Zbl 0960.14002

[16] Liu, Fu On positivity of Ehrhart polynomials, Recent trends in algebraic combinatorics (Assoc. Women Math. Ser.), Volume 16, Springer, Cham, 2019, pp. 189-237 | Article | MR 3969575 | Zbl 0869.52001

[17] McMullen, Peter Valuations and dissections, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993, pp. 933-988 | MR 1243000

[18] Pommersheim, James; Thomas, Hugh Cycles representing the Todd class of a toric variety, J. Amer. Math. Soc., Volume 17 (2004) no. 4, pp. 983-994 | Article | MR 2083474 | Zbl 0813.14039

[19] Sage Developers SageMath, the Sage Mathematics Software System (Version 9.2) (2020) (https://www.sagemath.org) | Article | MR 1415592 | Zbl 0885.14025

[20] Stanley, Richard P. Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages (With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin) | Article | MR 1676282 | Zbl 1266.14040

[21] Stanley, Richard P. Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, Cambridge, 2012, xiv+626 pages | Article | MR 2868112 | Zbl 0622.57014