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<p>The (<i>m</i>,&#xa0;<i>n</i>)-multiplihedron is a polytope whose faces correspond to <i>m</i>-painted <i>n</i>-trees, and whose oriented skeleton is the Hasse diagram of the rotation lattice on binary <i>m</i>-painted <i>n</i>-trees. Deleting certain inequalities from the facet description of the (<i>m</i>,&#xa0;<i>n</i>)-multiplihedron, we construct the (<i>m</i>,&#xa0;<i>n</i>)-Hochschild polytope whose faces correspond to <i>m</i>-lighted <i>n</i>-shades, and whose oriented skeleton is the Hasse diagram of the rotation lattice on unary <i>m</i>-lighted <i>n</i>-shades. Moreover, there is a natural shadow map from <i>m</i>-painted <i>n</i>-trees to <i>m</i>-lighted <i>n</i>-shades, which turns out to define a meet semilattice morphism of rotation lattices. In particular, when <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3120_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">\(m=1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, our Hochschild polytope is a deformed permutahedron whose oriented skeleton is the Hasse diagram of the Hochschild lattice.</p>

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Hochschild polytopes

  • 期刊论文

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  • 发表日期：

    2025年4月21日

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  • Vincent Pilaud,
  • Daria Poliakova

摘要

The (m, n)-multiplihedron is a polytope whose faces correspond to m-painted n-trees, and whose oriented skeleton is the Hasse diagram of the rotation lattice on binary m-painted n-trees. Deleting certain inequalities from the facet description of the (m, n)-multiplihedron, we construct the (m, n)-Hochschild polytope whose faces correspond to m-lighted n-shades, and whose oriented skeleton is the Hasse diagram of the rotation lattice on unary m-lighted n-shades. Moreover, there is a natural shadow map from m-painted n-trees to m-lighted n-shades, which turns out to define a meet semilattice morphism of rotation lattices. In particular, when \(m=1\) m = 1 , our Hochschild polytope is a deformed permutahedron whose oriented skeleton is the Hasse diagram of the Hochschild lattice.