Face numbers of Scarf complexesAnders BjornerKTHStockholm
November 17,

ABSTRACT


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 $f=(f_0,f_1,..., f_{d1})$ is the $f$vector of a certain triangulated $(d1)$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'' $I = \langle \mathbf{x}^{a}  \mathbf{x}^b  a,b\in N^{d+1} mbox{and} ab\in \{A\cdot y  y\in Z^d\} \rangle$. 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 $f_0,f_1, \dots, f_{\lfloor \frac{d3}{2} \rfloor}$ determine the other numbers via linear relations (confirming a conjecture of Scarf), and that there are additional nonlinear relations. In more precise (and more technical) terms, our analysis shows that $f$ is linearly determined by a certain $M$sequence $(g_0,g_1, \dots, g_{\lfloor \frac{d1}{2} \rfloor})$, namely the $g$vector of the $(d2)$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. 
Combinatorics Seminar, Mathematics Department, MIT, sara(atsign)math.mit.edu 

