Mathematics – Combinatorics
Scientific paper
2003-12-02
In "Integer Points in Polyhedra - Geometry, Number Theory, Algebra, Optimization", A. Barvinok, M. Beck, C. Haase, B. Reznick,
Mathematics
Combinatorics
This version has been accepted in "Proceedings of the Joint Summer Research Conference on Integer Points in Polyhedra" (Barvin
Scientific paper
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 $\Delta^3\times \Delta^3$.
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