Geodesic
Lattice polytopes from Schur and symmetric Grothendieck polynomials
Margaret Bayer1 ; Bennet Goeckner2 ; Su Ji Hong3 ; Tyrrell McAllister4 ; McCabe Olsen5 ; Casey Pinckney6 ; Julianne Vega7 ; Martha Yip7
1University of Kansas
2University of Washington
3Universtiy of Nebraska-Lincoln
4University of Wyoming
5The Ohio State University
6Colorado State University
7University of Kentucky
The electronic journal of combinatorics, Tome 28 (2021) no. 2
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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 $h^\ast$-polynomial. In this paper, we consider the Newton polytopes arising from two families of polynomials in algebraic combinatorics: Schur polynomials and inflated symmetric Grothendieck polynomials. 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 $h^\ast$-vector in the case of Schur polynomials.
DOI : 10.37236/9621
Classification : 52B20, 05E05

Margaret Bayer  1   ; Bennet Goeckner  2   ; Su Ji Hong  3   ; Tyrrell McAllister  4   ; McCabe Olsen  5   ; Casey Pinckney  6   ; Julianne Vega  7   ; Martha Yip  7

1 University of Kansas
2 University of Washington
3 Universtiy of Nebraska-Lincoln
4 University of Wyoming
5 The Ohio State University
6 Colorado State University
7 University of Kentucky 
@article{10_37236_9621,
     author = {Margaret Bayer and Bennet Goeckner and Su Ji Hong and Tyrrell McAllister and McCabe Olsen and Casey Pinckney and Julianne Vega and Martha Yip},
     title = {Lattice polytopes from {Schur} and symmetric {Grothendieck} polynomials},
     journal = {The electronic journal of combinatorics},
     year = {2021},
     volume = {28},
     number = {2},
     doi = {10.37236/9621},
     zbl = {1466.52018},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/9621/}
}
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%A Martha Yip
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Margaret Bayer; Bennet Goeckner; Su Ji Hong; Tyrrell McAllister; McCabe Olsen; Casey Pinckney; Julianne Vega; Martha Yip. Lattice polytopes from Schur and symmetric Grothendieck polynomials. The electronic journal of combinatorics, Tome 28 (2021) no. 2. doi: 10.37236/9621

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