An Algebraic Approach to Projective Uniqueness with an Application to Order Polytopes

Abstract

A combinatorial polytope P is said to be projectively unique if it has a single realization up to projective transformations. Projective uniqueness is a geometrically compelling property but is difficult to verify. In this paper, we merge two approaches to projective uniqueness in the literature. One is primarily geometric and is due to McMullen, who showed that certain natural operations on polytopes preserve projective uniqueness. The other is more algebraic and is due to Gouveia, Macchia, Thomas, and Wiebe. They use certain ideals associated to a polytope to verify a property called graphicality that implies projective uniqueness. In this paper, we show that McMullen’s operations preserve not only projective uniqueness but also graphicality. As an application, we show that large families of order polytopes are graphic and thus projectively unique.