Abstract <p>Motivated by categorical representation theory, we define the wiggly complex, whose vertices are arcs wiggling around <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3688_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$n+2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> points on a line, and whose faces are sets of wiggly arcs which are pairwise pointed and non-crossing. The wiggly complex is a <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3688_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$(2n-1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-dimensional pseudomanifold, whose facets are wiggly pseudotriangulations. We show that wiggly pseudotriangulations are in bijection with wiggly permutations, which are permutations of [2<i>n</i>] avoiding the patterns <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3688_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="144" /> </InlineMediaObject> <EquationSource Format="TEX">$(2j-1) \cdots i \cdots (2j)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mi>i</mi> <mo>⋯</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3688_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">$i < 2j-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo><</mo> <mn>2</mn> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3688_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="148" /> </InlineMediaObject> <EquationSource Format="TEX">$(2j) \cdots k \cdots (2j-1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo stretchy="false">)</mo> <mo>⋯</mo> <mi>k</mi> <mo>⋯</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3688_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$k > 2j$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>></mo> <mn>2</mn> <mi>j</mi> </mrow> </math></EquationSource> </InlineEquation>. These permutations define the wiggly lattice, an induced sublattice of the weak order. We then prove that the wiggly complex is isomorphic to the boundary complex of the polar of the wigglyhedron, for which we give explicit and simple vertex and facet descriptions. Interestingly, we observe that any Cambrian associahedron is normally equivalent to a well-chosen face of the wigglyhedron. Finally, we recall the correspondence of wiggly arcs with objects in a category, and we develop categorical criteria for a subset of wiggly arcs to form a face of the wiggly complex.</p> Graphical abstract

Wigglyhedra

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2025年5月9日

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Asilata Bapat,
Vincent Pilaud

摘要

Abstract

Motivated by categorical representation theory, we define the wiggly complex, whose vertices are arcs wiggling around $n+2$ $n + 2$ points on a line, and whose faces are sets of wiggly arcs which are pairwise pointed and non-crossing. The wiggly complex is a $(2n-1)$ $(2 n - 1)$ -dimensional pseudomanifold, whose facets are wiggly pseudotriangulations. We show that wiggly pseudotriangulations are in bijection with wiggly permutations, which are permutations of [2n] avoiding the patterns $(2j-1) \cdots i \cdots (2j)$ $(2 j - 1) \dots i \dots (2 j)$ for $i < 2j-1$ $i < 2 j - 1$ and $(2j) \cdots k \cdots (2j-1)$ $(2 j) \dots k \dots (2 j - 1)$ for $k > 2j$ $k > 2 j$ . These permutations define the wiggly lattice, an induced sublattice of the weak order. We then prove that the wiggly complex is isomorphic to the boundary complex of the polar of the wigglyhedron, for which we give explicit and simple vertex and facet descriptions. Interestingly, we observe that any Cambrian associahedron is normally equivalent to a well-chosen face of the wigglyhedron. Finally, we recall the correspondence of wiggly arcs with objects in a category, and we develop categorical criteria for a subset of wiggly arcs to form a face of the wiggly complex.

Graphical abstract