On Mirkovi\'c-Vilonen polytopes

Mirkovi\'c-Vilonen polytopes encode in a geometrical way the numerical data present in the Kashiwara crystal $B(\infty)$ of a semisimple group $G$. We retrieve these polytopes from the coproduct of the Hopf algebra $\mathscr O(N)$ of regular functions on a maximal unipotent subgroup $N$ of $G$. We bring attention to a remarkable behavior that the classical bases (dual canonical, dual semicanonical, Mirkovi\'c-Vilonen) of $\mathscr O(N)$ manifest with respect to the extremal points of these polytopes, which extends the crystal operations. This study leans on a notion of stability for graded bialgebras.