𝑚=2振幅面体和超单纯形:符号、簇、平铺、欧拉数

M. Parisi, M. Sherman-Bennett, Lauren Williams
{"title":"𝑚=2振幅面体和超单纯形:符号、簇、平铺、欧拉数","authors":"M. Parisi, M. Sherman-Bennett, Lauren Williams","doi":"10.1090/cams/23","DOIUrl":null,"url":null,"abstract":"<p>The hypersimplex <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Delta Subscript k plus 1 comma n\">\n <mml:semantics>\n <mml:msub>\n <mml:mi mathvariant=\"normal\">Δ<!-- Δ --></mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>k</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mi>n</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">\\Delta _{k+1,n}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is the image of the positive Grassmannian <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G r Subscript k plus 1 comma n Superscript greater-than-or-equal-to 0\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>G</mml:mi>\n <mml:msubsup>\n <mml:mi>r</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>k</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mi>n</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>≥<!-- ≥ --></mml:mo>\n <mml:mn>0</mml:mn>\n </mml:mrow>\n </mml:msubsup>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">Gr^{\\geq 0}_{k+1,n}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> under the moment map. It is a polytope of dimension <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n minus 1\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>n</mml:mi>\n <mml:mo>−<!-- − --></mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">n-1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> in <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R Superscript n\">\n <mml:semantics>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">R</mml:mi>\n </mml:mrow>\n <mml:mi>n</mml:mi>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {R}^n</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. Meanwhile, the amplituhedron <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper A Subscript n comma k comma 2 Baseline left-parenthesis upper Z right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">A</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>n</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>k</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mn>2</mml:mn>\n </mml:mrow>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>Z</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {A}_{n,k,2}(Z)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is the projection of the positive Grassmannian <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G r Subscript k comma n Superscript greater-than-or-equal-to 0\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>G</mml:mi>\n <mml:msubsup>\n <mml:mi>r</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>k</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>n</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>≥<!-- ≥ --></mml:mo>\n <mml:mn>0</mml:mn>\n </mml:mrow>\n </mml:msubsup>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">Gr^{\\geq 0}_{k,n}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> into the Grassmannian <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G r Subscript k comma k plus 2\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>G</mml:mi>\n <mml:msub>\n <mml:mi>r</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>k</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>k</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mn>2</mml:mn>\n </mml:mrow>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">Gr_{k,k+2}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> under a map <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Z overTilde\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mover>\n <mml:mi>Z</mml:mi>\n <mml:mo stretchy=\"false\">~<!-- ~ --></mml:mo>\n </mml:mover>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\tilde {Z}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> induced by a positive matrix <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Z element-of upper M a t Subscript n comma k plus 2 Superscript greater-than 0\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>Z</mml:mi>\n <mml:mo>∈<!-- ∈ --></mml:mo>\n <mml:mi>M</mml:mi>\n <mml:mi>a</mml:mi>\n <mml:msubsup>\n <mml:mi>t</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>n</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>k</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mn>2</mml:mn>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>></mml:mo>\n <mml:mn>0</mml:mn>\n </mml:mrow>\n </mml:msubsup>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">Z\\in Mat_{n,k+2}^{>0}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. Introduced in the context of <italic>scattering amplitudes</italic>, it is not a polytope, and has full dimension <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2 k\">\n <mml:semantics>\n <mml:mrow>\n <mml:mn>2</mml:mn>\n <mml:mi>k</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">2k</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> inside <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G r Subscript k comma k plus 2\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>G</mml:mi>\n <mml:msub>\n <mml:mi>r</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>k</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>k</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mn>2</mml:mn>\n </mml:mrow>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">Gr_{k,k+2}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. Nevertheless, there seem to be remarkable connections between these two objects via <italic>T-duality</italic>, as conjectured by Łukowski, Parisi, and Williams [Int. Math. Res. Not. (2023)]. In this paper we use ideas from oriented matroid theory, total positivity, and the geometry of the hypersimplex and positroid polytopes to obtain a deeper understanding of the amplituhedron. We show that the inequalities cutting out <italic>positroid polytopes</italic>—images of positroid cells of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G r Subscript k plus 1 comma n Superscript greater-than-or-equal-to 0\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>G</mml:mi>\n <mml:msubsup>\n <mml:mi>r</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>k</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mi>n</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>≥<!-- ≥ --></mml:mo>\n <mml:mn>0</mml:mn>\n </mml:mrow>\n </mml:msubsup>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">Gr^{\\geq 0}_{k+1,n}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> under the moment map—translate into sign conditions characterizing the T-dual <italic>Grasstopes</italic>—images of positroid cells of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G r Subscript k comma n Superscript greater-than-or-equal-to 0\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>G</mml:mi>\n <mml:msubsup>\n <mml:mi>r</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>k</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>n</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>≥<!-- ≥ --></mml:mo>\n <mml:mn>0</mml:mn>\n </mml:mrow>\n </mml:msubsup>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">Gr^{\\geq 0}_{k,n}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> under <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Z overTilde\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mover>\n <mml:mi>Z</mml:mi>\n <mml:mo stretchy=\"false\">~<!-- ~ --></mml:mo>\n </mml:mover>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\tilde {Z}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. Moreover, we subdivide the amplituhedron into <italic>chambers</italic>, just as the hypersimplex can be subdivided into simplices, with both chambers and simplices enumerated by the Eulerian numbers. We use these properties to prove ","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"46 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"The 𝑚=2 amplituhedron and the hypersimplex: Signs, clusters, tilings, Eulerian numbers\",\"authors\":\"M. Parisi, M. Sherman-Bennett, Lauren Williams\",\"doi\":\"10.1090/cams/23\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The hypersimplex <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal upper Delta Subscript k plus 1 comma n\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mi mathvariant=\\\"normal\\\">Δ<!-- Δ --></mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>k</mml:mi>\\n <mml:mo>+</mml:mo>\\n <mml:mn>1</mml:mn>\\n <mml:mo>,</mml:mo>\\n <mml:mi>n</mml:mi>\\n </mml:mrow>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\Delta _{k+1,n}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> is the image of the positive Grassmannian <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper G r Subscript k plus 1 comma n Superscript greater-than-or-equal-to 0\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>G</mml:mi>\\n <mml:msubsup>\\n <mml:mi>r</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>k</mml:mi>\\n <mml:mo>+</mml:mo>\\n <mml:mn>1</mml:mn>\\n <mml:mo>,</mml:mo>\\n <mml:mi>n</mml:mi>\\n </mml:mrow>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo>≥<!-- ≥ --></mml:mo>\\n <mml:mn>0</mml:mn>\\n </mml:mrow>\\n </mml:msubsup>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">Gr^{\\\\geq 0}_{k+1,n}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> under the moment map. It is a polytope of dimension <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"n minus 1\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>n</mml:mi>\\n <mml:mo>−<!-- − --></mml:mo>\\n <mml:mn>1</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">n-1</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> in <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper R Superscript n\\\">\\n <mml:semantics>\\n <mml:msup>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi>\\n </mml:mrow>\\n <mml:mi>n</mml:mi>\\n </mml:msup>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbb {R}^n</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. Meanwhile, the amplituhedron <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script upper A Subscript n comma k comma 2 Baseline left-parenthesis upper Z right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msub>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">A</mml:mi>\\n </mml:mrow>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>n</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>k</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mn>2</mml:mn>\\n </mml:mrow>\\n </mml:msub>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>Z</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathcal {A}_{n,k,2}(Z)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> is the projection of the positive Grassmannian <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper G r Subscript k comma n Superscript greater-than-or-equal-to 0\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>G</mml:mi>\\n <mml:msubsup>\\n <mml:mi>r</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>k</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>n</mml:mi>\\n </mml:mrow>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo>≥<!-- ≥ --></mml:mo>\\n <mml:mn>0</mml:mn>\\n </mml:mrow>\\n </mml:msubsup>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">Gr^{\\\\geq 0}_{k,n}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> into the Grassmannian <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper G r Subscript k comma k plus 2\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>G</mml:mi>\\n <mml:msub>\\n <mml:mi>r</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>k</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>k</mml:mi>\\n <mml:mo>+</mml:mo>\\n <mml:mn>2</mml:mn>\\n </mml:mrow>\\n </mml:msub>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">Gr_{k,k+2}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> under a map <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper Z overTilde\\\">\\n <mml:semantics>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mover>\\n <mml:mi>Z</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">~<!-- ~ --></mml:mo>\\n </mml:mover>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\tilde {Z}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> induced by a positive matrix <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper Z element-of upper M a t Subscript n comma k plus 2 Superscript greater-than 0\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>Z</mml:mi>\\n <mml:mo>∈<!-- ∈ --></mml:mo>\\n <mml:mi>M</mml:mi>\\n <mml:mi>a</mml:mi>\\n <mml:msubsup>\\n <mml:mi>t</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>n</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>k</mml:mi>\\n <mml:mo>+</mml:mo>\\n <mml:mn>2</mml:mn>\\n </mml:mrow>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo>></mml:mo>\\n <mml:mn>0</mml:mn>\\n </mml:mrow>\\n </mml:msubsup>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">Z\\\\in Mat_{n,k+2}^{>0}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. Introduced in the context of <italic>scattering amplitudes</italic>, it is not a polytope, and has full dimension <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"2 k\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mn>2</mml:mn>\\n <mml:mi>k</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">2k</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> inside <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper G r Subscript k comma k plus 2\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>G</mml:mi>\\n <mml:msub>\\n <mml:mi>r</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>k</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>k</mml:mi>\\n <mml:mo>+</mml:mo>\\n <mml:mn>2</mml:mn>\\n </mml:mrow>\\n </mml:msub>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">Gr_{k,k+2}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. Nevertheless, there seem to be remarkable connections between these two objects via <italic>T-duality</italic>, as conjectured by Łukowski, Parisi, and Williams [Int. Math. Res. Not. (2023)]. In this paper we use ideas from oriented matroid theory, total positivity, and the geometry of the hypersimplex and positroid polytopes to obtain a deeper understanding of the amplituhedron. We show that the inequalities cutting out <italic>positroid polytopes</italic>—images of positroid cells of <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper G r Subscript k plus 1 comma n Superscript greater-than-or-equal-to 0\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>G</mml:mi>\\n <mml:msubsup>\\n <mml:mi>r</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>k</mml:mi>\\n <mml:mo>+</mml:mo>\\n <mml:mn>1</mml:mn>\\n <mml:mo>,</mml:mo>\\n <mml:mi>n</mml:mi>\\n </mml:mrow>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo>≥<!-- ≥ --></mml:mo>\\n <mml:mn>0</mml:mn>\\n </mml:mrow>\\n </mml:msubsup>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">Gr^{\\\\geq 0}_{k+1,n}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> under the moment map—translate into sign conditions characterizing the T-dual <italic>Grasstopes</italic>—images of positroid cells of <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper G r Subscript k comma n Superscript greater-than-or-equal-to 0\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>G</mml:mi>\\n <mml:msubsup>\\n <mml:mi>r</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>k</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>n</mml:mi>\\n </mml:mrow>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo>≥<!-- ≥ --></mml:mo>\\n <mml:mn>0</mml:mn>\\n </mml:mrow>\\n </mml:msubsup>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">Gr^{\\\\geq 0}_{k,n}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> under <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper Z overTilde\\\">\\n <mml:semantics>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mover>\\n <mml:mi>Z</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">~<!-- ~ --></mml:mo>\\n </mml:mover>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\tilde {Z}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. Moreover, we subdivide the amplituhedron into <italic>chambers</italic>, just as the hypersimplex can be subdivided into simplices, with both chambers and simplices enumerated by the Eulerian numbers. We use these properties to prove \",\"PeriodicalId\":285678,\"journal\":{\"name\":\"Communications of the American Mathematical Society\",\"volume\":\"46 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-07-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications of the American Mathematical Society\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/cams/23\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications of the American Mathematical Society","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/cams/23","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 2

摘要

《hypersimplex Δ k + 1 , n \ 三角洲的{k + 1, n}是阳性Grassmannian形象》 G r k + 1 , n ≥ 0 Gr ^ {\ geq 0} {k + 1, n}下的《地图的时刻。这是a的polytope维度 n−1 n-1 in R n \ mathbb {R) ^ n。Meanwhile, the amplituhedron A n , k , 2 ( Z ) \ mathcal {A} {n, k, 2}的投射》(Z)是阳性Grassmannian G r k , n ≥ 0 Gr ^ {\ geq 0} {k, n}》的Grassmannian G r k ,k + 2 Gr_ {k, k + 2下的a地图 Z ~ \ 蒂尔德{Z} induced by a阳性矩阵 公元Z∈a t n , k + 2 > 0 Z \在Mat_ {n, k + 2) ^{> 0}。在散射振幅的背景下进行介绍,它不是一个多边形,里面有一个2k的尺寸在Gr r k,k+2 G {k,k+2}。Nevertheless,那里似乎成为非凡的connections这些通过T-duality两个物体之间,美国conjectured由Łukowski,帕里和威廉姆斯(Int)。数学。Res音符。(2023)。在这篇文章中,我们用的是来自东方matroid理论的概念,全正常性,以及高度对称和波利托对偶的几何知识。我们那个节目《不平等卡特房positroid polytopes——images of positroid细胞of G r k + 1 , n ≥ 0 Gr ^ {\ geq 0} {k + 1, n}下的《地图的时刻——翻译进入签约条件characterizing T-dual Grasstopes——images of positroid细胞of G r k ,n ≥ 0 Gr ^ {\ geq 0} {k, n}下的 Z ~ \ 蒂尔德{Z}。更重要的是,我们把振幅移植到钱伯斯,就像超硬化剂可以移植到钱伯斯,由长老会的数字修剪。我们用这些属性来证明
本文章由计算机程序翻译,如有差异,请以英文原文为准。
The 𝑚=2 amplituhedron and the hypersimplex: Signs, clusters, tilings, Eulerian numbers

The hypersimplex Δ k + 1 , n \Delta _{k+1,n} is the image of the positive Grassmannian G r k + 1 , n 0 Gr^{\geq 0}_{k+1,n} under the moment map. It is a polytope of dimension n 1 n-1 in R n \mathbb {R}^n . Meanwhile, the amplituhedron A n , k , 2 ( Z ) \mathcal {A}_{n,k,2}(Z) is the projection of the positive Grassmannian G r k , n 0 Gr^{\geq 0}_{k,n} into the Grassmannian G r k , k + 2 Gr_{k,k+2} under a map Z ~ \tilde {Z} induced by a positive matrix Z M a t n , k + 2 > 0 Z\in Mat_{n,k+2}^{>0} . Introduced in the context of scattering amplitudes, it is not a polytope, and has full dimension 2 k 2k inside G r k , k + 2 Gr_{k,k+2} . Nevertheless, there seem to be remarkable connections between these two objects via T-duality, as conjectured by Łukowski, Parisi, and Williams [Int. Math. Res. Not. (2023)]. In this paper we use ideas from oriented matroid theory, total positivity, and the geometry of the hypersimplex and positroid polytopes to obtain a deeper understanding of the amplituhedron. We show that the inequalities cutting out positroid polytopes—images of positroid cells of G r k + 1 , n 0 Gr^{\geq 0}_{k+1,n} under the moment map—translate into sign conditions characterizing the T-dual Grasstopes—images of positroid cells of G r k , n 0 Gr^{\geq 0}_{k,n} under Z ~ \tilde {Z} . Moreover, we subdivide the amplituhedron into chambers, just as the hypersimplex can be subdivided into simplices, with both chambers and simplices enumerated by the Eulerian numbers. We use these properties to prove

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