An algebraic approach to projective uniqueness with an application to order polytopes. (English) Zbl 1487.13055

Recall that a combinatorial polytope is said to be projectively unique if it has a single realization up to projective transformations. Note that the projective uniqueness is difficult to verify even though it is geometrically captivating.
A natural geometric approach to studying the projective uniqueness of the combinatorial polytopes is to investigate operations on polytopes that preserve projective uniqueness. P. McMullen described the operations on polytopes that preserve the projective uniqueness in [Discrete Math. 14, 347–358 (1976; Zbl 0319.52010)]. On the other hand, an algebraic approach to studying the projective uniqueness has appeared recently. J. Gouveia et al. [J. Comb. Theory, Ser. A 145, 184–226 (2017; Zbl 1360.52006)] introduced the slack ideals of polytopes (to study its positive semidefinite lifts) and J. Gouveia et al. [SIAM J. Discrete Math. 33, No. 3, 1637–1653 (2019; Zbl 1423.52032); J. Pure Appl. Algebra 224, No. 5, Article ID 106229, 14 p. (2020; Zbl 1434.52015)] applied the slack ideals of polytopes to give an algebraic criterion for projective uniqueness. A subclass of projectively unique polytopes was defined, the graphic polytopes, for which one has an algebraic certificate of projective uniqueness.
In this paper, the authors merge two approaches to projective uniqueness in the literature. They prove that McMullen’s operations preserve not only projective uniqueness but also graphicality.
As an application, they show that large families of order polytopes are graphic, and therefore projectively unique. In detail, they prove that order polytopes from finite ranked posets with no 3-antichain are graphic, and thus projectively unique.


13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
52B12 Special polytopes (linear programming, centrally symmetric, etc.)
06A07 Combinatorics of partially ordered sets
52C25 Rigidity and flexibility of structures (aspects of discrete geometry)


2L_enum; SlackIdeals
Full Text: DOI arXiv


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