The Cayley trick and triangulations of products of simplices

TL;DR

This paper uses the Cayley Trick to analyze triangulations of products of simplices, revealing their growth rates, connections to lozenge tilings, and the connectedness of their flip graph, with exact counts for small cases.

Contribution

It establishes growth rates of triangulations, relates them to lozenge tilings for l=2, and proves the connectivity of the triangulation flip graph, with explicit enumeration for small cases.

Findings

Number of regular triangulations grows as k^{Theta(k)}.

Number of non-regular triangulations grows as 2^{Omega(k^2)}.

The set of all triangulations is connected under bistellar flips.

Abstract

We use the Cayley Trick to study polyhedral subdivisions of the product of two simplices. For arbitrary (fixed) $l$, we show that the numbers of regular and non-regular triangulations of $\Delta^l\times\Delta^k$ grow, respectively, as $k^{\Theta(k)}$ and $2^{\Omega(k^2)}$. For the special case of $l=2$, we relate triangulations to certain class of lozenge tilings. This allows us to compute the exact number of triangulations up to $k=15$, show that the number grows as $e^{\beta k^2/2 + o(k^2)}$ where $\beta\simeq 0.32309594$ and prove that the set of all triangulations is connected under geometric bistellar flips. The latter has as a corollary that the toric Hilbert scheme of the determinantal ideal of $2\times 2$ minors of a $3\times k$ matrix is connected, for every $k$. We include ``Cayley Trick pictures'' of all the triangulations of $\Delta^2\times \Delta^2$ and $\Delta^2\times \Delta^3$, as well as one non-regular triangulation of $\Delta^2\times \Delta^5$ and one of $\Delta^3\times \Delta^3$.