The Cayley trick and triangulations of products of simplices

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 $\Delta^3\times \Delta^3$.