The Combinatorics of Triangulations of Products of Two Simplices

We study the combinatorial structure of triangulations of the Cartesian product of two standard simplices, a seemingly simple object that is deeply connected to many objects spanning across polyhedral geometry, tropical geometry, topology, algebraic geometry, and matroid theory. In the first part, we extend our current understanding in the case where one of the two simplices has dimension two or three. In the second part, we establish several new cryptomorphisms between triangulations and related objects, addressing some gaps in the current literature. Finally, we study a generalization where the triangulated polytope is a sub-polytope of the product of two simplices, and generalize the related combinatorial characterizations accordingly.

For information about the Richard P. Stanley Seminar in Combinatorics, visit… https://math.mit.edu/combin/