  We give a necessary and sufficient condition for extremality of a
supermodular function based on its min-representation by means of (vertices of)
the corresponding core polytope. The condition leads to solving a certain
simple linear equation system determined by the combinatorial core structure.
Our result allows us to characterize indecomposability in the class of
generalized permutohedra. We provide an in-depth comparison between our result
and the description of extremality in the supermodular/submodular cone achieved
by other researchers.
