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arXiv · 2401.10340

On Mirkovi\'c-Vilonen polytopes

Pierre Baumann
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Abstract

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.

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Pierre Baumann. 2024-01-18. On Mirkovi\'c-Vilonen polytopes. https://arxiv.org/abs/2401.10340

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