TY - JOUR

T1 - Lefschetz properties of balanced 3-polytopes

AU - Cook, David

AU - Juhnke-Kubitzke, Martina

AU - Murai, Satoshi

AU - Nevo, Eran

N1 - Funding Information:
The research of the second author was partially supported by the German Research Council, DFG-GRK 1916. The research of the third author was partially supported by KAKENHI16K05102. The research of the fourth author was partially supported by the Israel Science Foundation, grant No. ISF-1695/15.
Publisher Copyright:
Copyright © 2018 Rocky Mountain Mathematics Consortium.

PY - 2018

Y1 - 2018

N2 - In this paper, we study Lefschetz properties of Artinian reductions of Stanley-Reisner rings of balanced simplicial 3-polytopes. A (d − 1)-dimensional simplicial complex is said to be balanced if its graph is d-colorable. If a simplicial complex is balanced, then its Stanley-Reisner ring has a special system of parameters induced by the coloring. We prove that the Artinian reduction of the Stanley-Reisner ring of a balanced simplicial 3-polytope with respect to this special system of parameters has the strong Lefschetz property if the characteristic of the base field is not two or three. Moreover, we characterize (2, 1)-balanced simplicial polytopes, i.e., polytopes with exactly one red vertex and two blue vertices in each facet, such that an analogous property holds. In fact, we show that this is the case if and only if the induced graph on the blue vertices satisfies a Laman-type combinatorial condition.

AB - In this paper, we study Lefschetz properties of Artinian reductions of Stanley-Reisner rings of balanced simplicial 3-polytopes. A (d − 1)-dimensional simplicial complex is said to be balanced if its graph is d-colorable. If a simplicial complex is balanced, then its Stanley-Reisner ring has a special system of parameters induced by the coloring. We prove that the Artinian reduction of the Stanley-Reisner ring of a balanced simplicial 3-polytope with respect to this special system of parameters has the strong Lefschetz property if the characteristic of the base field is not two or three. Moreover, we characterize (2, 1)-balanced simplicial polytopes, i.e., polytopes with exactly one red vertex and two blue vertices in each facet, such that an analogous property holds. In fact, we show that this is the case if and only if the induced graph on the blue vertices satisfies a Laman-type combinatorial condition.

KW - And phrases. Stanley-Riesner rings

KW - Balanced complexes

KW - Laman graphs

KW - Lefschetz properties

KW - Simplicial polytopes

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U2 - 10.1216/RMJ-2018-48-3-769

DO - 10.1216/RMJ-2018-48-3-769

M3 - Article

AN - SCOPUS:85082349501

VL - 48

SP - 769

EP - 790

JO - Rocky Mountain Journal of Mathematics

JF - Rocky Mountain Journal of Mathematics

SN - 0035-7596

IS - 3

ER -