Mirković–Vilonen polytopes encode in a geometrical way the numerical data present in the Kashiwara crystal <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10468_2025_10317_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$B(\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mo stretchy="false">(</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of a semisimple group G. We retrieve these polytopes from the coproduct of the Hopf algebra <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10468_2025_10317_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathscr {O}(N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10468_2025_10317_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathscr {O}(N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> 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.

On Mirković–Vilonen Polytopes

期刊论文
发表日期：

2025年2月4日

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Pierre Baumann

摘要

Mirković–Vilonen polytopes encode in a geometrical way the numerical data present in the Kashiwara crystal $B(\infty )$ $B (\infty)$ of a semisimple group G. We retrieve these polytopes from the coproduct of the Hopf algebra $\mathscr {O}(N)$ $O (N)$ 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 $\mathscr {O}(N)$ $O (N)$ 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.