Theory Update 82

These days Arkani-Hamed raves about matroid polytopes, which he has been learning about from the mathematicians. Matroid polytopes are generalised permutohedra, a la Postnikov.

Recall that a permutohedron in dimension $n$ is specified by the (convex hull of all) permutations of the coordinate $(1,2,3,4,\cdots,n+1)$. Similarly, a generalised permutohedron $P_{n}$ is specified by a list $\{ z_{i} \}$ of real numbers. The Minkowski sum of two polytopes is, by definition, the collection of all points $p + q$, where $p$ is in one polytope and $q$ in the other. Under Minkowski sum, polytopes have the simplest possible functorial addition rule

$P_n(\{ z_{i} \}) + P_n(\{ t_{i} \}) = P_n (\{ z_{i} + t_{i} \})$.

Every generalised permutohedron may be written as a signed sum of simplices.