Face numbers of Scarf complexes

Anders Bjorner


November 17,
refreshments at 3:45pm


Let $A$ be a $(d+1) \times d$ real matrix whose row vectors positively span $R^d$ and which is generic in the sense of Bárány and Scarf. Such a matrix determines a certain infinite $d$-dimensional simplicial complex $\Sigma$, on which the group $Z^d$ acts with finitely many orbits. Let $f_i$ be the number of orbits of $(i+1)$-simplices of $\Sigma$. The sequence $f=(f_0,f_1,..., f_{d-1})$ is the $f$-vector of a certain triangulated $(d-1)$-ball $T$ embedded in $\Sigma$. When $A$ has integer entries it is also, as shown by work of Peeva and Sturmfels, the sequence of Betti numbers of the minimal free resolution of $k[x_1,...,x_{d+1}]/I$, where $I$ is the ``lattice ideal'' $I = \langle \mathbf{x}^{a} - \mathbf{x}^b | a,b\in N^{d+1} mbox{and} a-b\in \{A\cdot y | y\in Z^d\} \rangle$.

The talk will give an introduction to the construction of such ``Scarf complexes'' $\Sigma$ and $T$ and their motivation. No previous familiarity will be assumed.

Then we will discuss relations among the numbers $f_i$. It is shown that $f_0,f_1, \dots, f_{\lfloor \frac{d-3}{2} \rfloor}$ determine the other numbers via linear relations (confirming a conjecture of Scarf), and that there are additional non-linear relations. In more precise (and more technical) terms, our analysis shows that $f$ is linearly determined by a certain $M$-sequence $(g_0,g_1, \dots, g_{\lfloor \frac{d-1}{2} \rfloor})$, namely the $g$-vector of the $(d-2)$-sphere bounding $T$. Although $T$ is in general not a cone over its boundary, it turns out that its $f$-vector behaves as if it were.

Speaker's Contact Info: bjorner(at-sign)math.kth.se

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