Lattice polytopes from schur and symmetric grothendieck polynomials

Margaret Bayer; Bennet Goeckner; Su Ji Hong; Tyrrell McAllister; McCabe Olsen; Casey Pinckney; Julianne Vega; Martha Yip

doi:10.37236/9621

Lattice polytopes from schur and symmetric grothendieck polynomials

Margaret Bayer, Bennet Goeckner, Su Ji Hong, Tyrrell McAllister, McCabe Olsen, Casey Pinckney, Julianne Vega, Martha Yip

Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

Given a family of lattice polytopes, two common questions in Ehrhart Theory are determining when a polytope has the integer decomposition property and determining when a polytope is reflexive. While these properties are of independent interest, the confluence of these properties is a source of active investigation due to conjectures regarding the unimodality of the h^∗-polynomial. In this paper, we consider the Newton polytopes arising from two families of polynomials in algebraic combinatorics: Schur polynomials and inflated symmetric Grothendieck polynomi-als. In both cases, we prove that these polytopes have the integer decomposition property by using the fact that both families of polynomials have saturated Newton polytope. Furthermore, in both cases, we provide a complete characterization of when these polytopes are reflexive. We conclude with some explicit formulas and unimodality implications of the h^∗-vector in the case of Schur polynomials.

Original language	English
Article number	P2.45
Journal	Electronic Journal of Combinatorics
Volume	28
Issue number	2
DOIs	https://doi.org/10.37236/9621
State	Published - 2021

Bibliographical note

Publisher Copyright:
© The authors.

Funding

∗Partially supported by University of Kansas General Research Fund. †Partially supported by AMS-Simons Travel Grant. ‡Partially supported by Simons Collaboration Grant. Partially supported by University of Kansas General Research Fund. Partially supported by AMS-Simons Travel Grant. Partially supported by Simons Collaboration Grant. The authors would like to thank Federico Castillo and Semin Yoo for helpful discussions and contributions to the early stages of the project. We would also like to thank Michel Marcus and Avery St. Dizier for helpful comments on a previous version of the paper. This work was completed in part at the 2019 Graduate Research Workshop in Combinatorics, which was supported in part by NSF grant #1923238, NSA grant #H98230-18-1-0017, a generous award from the Combinatorics Foundation, and Simons Foundation Collaboration Grants #426971 (to M. Ferrara) and #315347 (to J. Martin). This work was completed in part at the 2019 Graduate Research Workshop in Combinatorics, which was supported in part by NSF grant #1923238, NSA grant #H98230-18-1-0017, a generous award from the Combinatorics Foundation, and Simons Foundation Collaboration Grants #426971 (to M. Ferrara) and #315347 (to J. Martin).

Funders	Funder number
AMS-Simons
Combinatorics Foundation
National Science Foundation (NSF)	1923238
Simons Foundation	315347, 426971
University of Kansas and University of Kansas Cancer Center
National Security Agency	98230-18-1-0017

ASJC Scopus subject areas

Theoretical Computer Science
Geometry and Topology
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics
Applied Mathematics

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10.37236/9621

Cite this

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abstract = "Given a family of lattice polytopes, two common questions in Ehrhart Theory are determining when a polytope has the integer decomposition property and determining when a polytope is reflexive. While these properties are of independent interest, the confluence of these properties is a source of active investigation due to conjectures regarding the unimodality of the h∗-polynomial. In this paper, we consider the Newton polytopes arising from two families of polynomials in algebraic combinatorics: Schur polynomials and inflated symmetric Grothendieck polynomi-als. In both cases, we prove that these polytopes have the integer decomposition property by using the fact that both families of polynomials have saturated Newton polytope. Furthermore, in both cases, we provide a complete characterization of when these polytopes are reflexive. We conclude with some explicit formulas and unimodality implications of the h∗-vector in the case of Schur polynomials.",

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AU - Bayer, Margaret

AU - Goeckner, Bennet

AU - Hong, Su Ji

AU - McAllister, Tyrrell

AU - Olsen, McCabe

AU - Pinckney, Casey

AU - Vega, Julianne

AU - Yip, Martha

PY - 2021

Y1 - 2021

N2 - Given a family of lattice polytopes, two common questions in Ehrhart Theory are determining when a polytope has the integer decomposition property and determining when a polytope is reflexive. While these properties are of independent interest, the confluence of these properties is a source of active investigation due to conjectures regarding the unimodality of the h∗-polynomial. In this paper, we consider the Newton polytopes arising from two families of polynomials in algebraic combinatorics: Schur polynomials and inflated symmetric Grothendieck polynomi-als. In both cases, we prove that these polytopes have the integer decomposition property by using the fact that both families of polynomials have saturated Newton polytope. Furthermore, in both cases, we provide a complete characterization of when these polytopes are reflexive. We conclude with some explicit formulas and unimodality implications of the h∗-vector in the case of Schur polynomials.

AB - Given a family of lattice polytopes, two common questions in Ehrhart Theory are determining when a polytope has the integer decomposition property and determining when a polytope is reflexive. While these properties are of independent interest, the confluence of these properties is a source of active investigation due to conjectures regarding the unimodality of the h∗-polynomial. In this paper, we consider the Newton polytopes arising from two families of polynomials in algebraic combinatorics: Schur polynomials and inflated symmetric Grothendieck polynomi-als. In both cases, we prove that these polytopes have the integer decomposition property by using the fact that both families of polynomials have saturated Newton polytope. Furthermore, in both cases, we provide a complete characterization of when these polytopes are reflexive. We conclude with some explicit formulas and unimodality implications of the h∗-vector in the case of Schur polynomials.

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Lattice polytopes from schur and symmetric grothendieck polynomials

Abstract

Bibliographical note

Funding

ASJC Scopus subject areas

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