On Mirković–Vilonen Polytopes

Abstract

Mirković–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, Mirković–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.