ALGEBRAIC COMBINATORICS

no. 3
On the Todd class of the permutohedral variety
Castillo, Federico1; Liu, Fu2
1 Max Planck Institutefor Mathematics in the Sciences Inselstr. 22 04103 Leipzig Germany
2 Department of Mathematics University of California, Davis One Shields Avenue, Davis, CA 95616 USA.
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≥24.

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

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.157
Classification: 52B20, 14M25
Keywords: Ehrhart polynomials, generalized permutohedra, Berline–Vergne construction
Castillo, Federico 1; Liu, Fu 2

1 Max Planck Institutefor Mathematics in the Sciences Inselstr. 22 04103 Leipzig Germany
2 Department of Mathematics University of California, Davis One Shields Avenue, Davis, CA 95616 USA.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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 | DOI | MR | Zbl

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

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

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

[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 | DOI | MR | Zbl

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

[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 | DOI | MR

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

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

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

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

[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) | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

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

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

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

[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) | DOI | MR | Zbl

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

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