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

Bogart, Tristram; Gouveia, João; Torres, Juan Camilo

Discrete Comput. Geom. 67, No. 2, 462-491 (2022).

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.

Reviewer: Jaewoo Jung (Atlanta)

MSC:

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)

Keywords:

McMullen’s operations; order polytopes; projectively unique polytopes; slack ideals

Citations:

Zbl 0319.52010; Zbl 1360.52006; Zbl 1423.52032; Zbl 1434.52015

Software:

2L_enum; SlackIdeals

References:

[1]	Adiprasito, KA; Ziegler, GM, Many projectively unique polytopes, Invent. Math., 199, 3, 581-652 (2015) · Zbl 1339.52011
[2]	Aprile, M.; Cevallos, A.; Faenza, Y., On \(2\)-level polytopes arising in combinatorial settings, SIAM J. Discrete Math., 32, 3, 1857-1886 (2018) · Zbl 1395.52018
[3]	Bohn, A.; Faenza, Y.; Fiorini, S.; Fisikopoulos, V.; Macchia, M.; Pashkovich, K., Enumeration of \(2\)-level polytopes, Math. Program. Comput., 11, 1, 173-210 (2019) · Zbl 1414.05023
[4]	Gouveia, J.; Grappe, R.; Kaibel, V.; Pashkovich, K.; Robinson, RZ; Thomas, RR, Which nonnegative matrices are slack matrices?, Linear Algebra Appl., 439, 10, 2921-2933 (2013) · Zbl 1283.15103
[5]	Gouveia, J.; Macchia, A.; Thomas, RR; Wiebe, A., The slack realization space of a polytope, SIAM J. Discrete Math., 33, 3, 1637-1653 (2019) · Zbl 1423.52032
[6]	Gouveia, J., Macchia, A., Thomas, R.R., Wiebe, A.: Projectively unique polytopes and toric slack ideals. J. Pure Appl. Algebra 224(5), # 106229 (2020) · Zbl 1434.52015
[7]	Gouveia, J.; Pashkovich, K.; Robinson, RZ; Thomas, RR, Four-dimensional polytopes of minimum positive semidefinite rank, J. Comb. Theory Ser. A, 145, 184-226 (2017) · Zbl 1360.52006
[8]	Grünbaum, B., Convex Polytopes (2003), New York: Springer, New York · Zbl 1024.52001
[9]	Macchia, A., Wiebe, A.: Slack ideals in Macaulay2 (2020). arXiv:2003.07382
[10]	McMullen, P., Constructions for projectively unique polytopes, Discrete Math., 14, 4, 347-358 (1976) · Zbl 0319.52010
[11]	Perles, MA; Shephard, GC, A construction for projectively unique polytopes, Geometriae Dedicata, 3, 3, 357-363 (1974) · Zbl 0299.52004
[12]	Richter-Gebert, J.: Realization Spaces of Polytopes. Lecture Notes in Mathematics, vol. 1643. Springer, Berlin (1996) · Zbl 0866.52009
[13]	Stanley, RP, Two poset polytopes, Discrete Comput. Geom., 1, 1, 9-23 (1986) · Zbl 0595.52008
[14]	Stanley, R.P.: Enumerative Combinatorics. Vol. 1. Cambridge Studies in Advanced Mathematics, vol. 49. Cambridge University Press, Cambridge (2012)
[15]	Steinitz, E.: Polyeder und Raumeinteilungen. In: Encyklopädie der Mathematischen Wissenschaften, vol. 3-1-2, # 12 (1922)
[16]	Swanson, I.; Walker, RM, Tensor-multinomial sums of ideals: primary decompositions and persistence of associated primes, Proc. Am. Math. Soc., 147, 12, 5071-5082 (2019) · Zbl 1436.13018
[17]	Wiebe, A.: Realization Spaces of Polytopes and Matroids. PhD thesis, University of Washington (2019)
[18]	Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995) · Zbl 0823.52002