On Mirković-Vilonen polytopes

2024/01/18 by Baumann, Pierre
#22E46 (Primary) 14M15 (Secondary) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2401.10340

Abstract

Mirković-Vilonen polytopes encode in a geometrical way the numerical data present in the Kashiwara crystal B(∞) 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.

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