CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> D6 in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> D6 </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/10273/#Item_1" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <blockquote> <p>This entry is about items in the <a class="existingWikiWord" href="/nlab/show/ADE-classification">ADE-classification</a> labeled by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mn>6</mn></mrow><annotation encoding="application/x-tex">D6</annotation></semantics></math>. For the <a class="existingWikiWord" href="/nlab/show/D6-brane">D6-brane</a>, see there.</p> </blockquote> <hr /> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="mathematics">Mathematics</h4> <div class="hide"><div> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/mathematics">mathematics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/math+resources">math resources</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/history+of+mathematics">history of mathematics</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/foundations">Structural Foundations</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/logic">logic</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/internal+language">internal language</a></li> <li><a class="existingWikiWord" href="/nlab/show/classical+mathematics">classical mathematics</a></li> <li><a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a></li> <li><a class="existingWikiWord" href="/nlab/show/predicative+mathematics">predicative mathematics</a></li> <li><a href="http://ncatlab.org/nlab/list/foundational+axiom">category:foundational axiom</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/structural+set+theory">structural set theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Categories+and+Sheaves">Categories and Sheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Sheaves+in+Geometry+and+Logic">Sheaves in Geometry and Logic</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher topos theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">(∞,1)-topos theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/models+for+%E2%88%9E-stack+%28%E2%88%9E%2C1%29-toposes">models for ∞-stack (∞,1)-toposes</a></li> <li><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a></li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topology+and+geometry">Topology and Geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> (general list), <a class="existingWikiWord" href="/nlab/show/topology">topology</a> (general list)</li> <li><a class="existingWikiWord" href="/nlab/show/general+topology">general topology</a></li> <li><a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a></li> <li><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a></li> <li><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a></li> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+algebraic+geometry">noncommutative algebraic geometry</a></li> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a> (general flavour)</li> <li><a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra">Algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+theory">group theory</a>, <a class="existingWikiWord" href="/nlab/show/ring+theory">ring theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+approaches+to+differential+calculus">algebraic approaches to differential calculus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/counterexamples+in+algebra">counterexamples in algebra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/analysis">analysis</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/nonstandard+analysis">nonstandard analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/operator+algebras">operator algebras</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fourier+transform">Fourier transform</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/infinity-Lie+theory+-+contents">higher Lie theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/probability+theory">probability theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+mathematics">discrete mathematics</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#related_concepts'>Related concepts</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In the <a class="existingWikiWord" href="/nlab/show/ADE-classification">ADE-classification</a>, the items labeled <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mn>6</mn></mrow><annotation encoding="application/x-tex">D6</annotation></semantics></math> include the following:</p> <ol> <li> <p>as <a class="existingWikiWord" href="/nlab/show/finite+subgroups+of+SO%283%29">finite subgroups of SO(3)</a>:</p> <p>the <a class="existingWikiWord" href="/nlab/show/dihedral+group+of+order+8">dihedral group of order 8</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔻</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{D}_8</annotation></semantics></math></p> </li> <li> <p>as <a class="existingWikiWord" href="/nlab/show/finite+subgroups+of+SU%282%29">finite subgroups of SU(2)</a>:</p> <p>the <a class="existingWikiWord" href="/nlab/show/binary+dihedral+group+of+order+16">binary dihedral group of order 16</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><msub><mi>D</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">2 D_8</annotation></semantics></math></p> </li> <li> <p>as <a class="existingWikiWord" href="/nlab/show/simple+Lie+groups">simple Lie groups</a>: the <a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a>/<a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a> in 12 dimensions</p> <p><a class="existingWikiWord" href="/nlab/show/SO%2812%29">SO(12)</a>, <a class="existingWikiWord" href="/nlab/show/Spin%2812%29">Spin(12)</a></p> </li> <li> <p>as a <a class="existingWikiWord" href="/nlab/show/Dynkin+diagram">Dynkin diagram</a>/<a class="existingWikiWord" href="/nlab/show/Dynkin+quiver">Dynkin quiver</a>:</p> </li> </ol> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="107.016" height="56.901" viewBox="0 0 107.016 56.901"> <defs> <clipPath id="H23pwce3oeQ5znPcs93QPCghKbw=-clip-0"> <path clip-rule="nonzero" d="M 94 0 L 107.015625 0 L 107.015625 13 L 94 13 Z M 94 0 "></path> </clipPath> <clipPath id="H23pwce3oeQ5znPcs93QPCghKbw=-clip-1"> <path clip-rule="nonzero" d="M 94 44 L 107.015625 44 L 107.015625 56.902344 L 94 56.902344 Z M 94 44 "></path> </clipPath> </defs> <path fill-rule="nonzero" fill="rgb(0%, 0%, 0%)" fill-opacity="1" stroke-width="0.3985" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 2.836344 0.00178125 C 2.836344 1.564281 1.566812 2.833812 0.00040625 2.833812 C -1.566 2.833812 -2.835531 1.564281 -2.835531 0.00178125 C -2.835531 -1.564625 -1.566 -2.834156 0.00040625 -2.834156 C 1.566812 -2.834156 2.836344 -1.564625 2.836344 0.00178125 Z M 2.836344 0.00178125 " transform="matrix(1, 0, 0, -1, 88.941, 28.451)"></path> <path fill-rule="nonzero" fill="rgb(0%, 0%, 0%)" fill-opacity="1" d="M 105.949219 3.902344 C 105.949219 2.335938 104.679688 1.066406 103.113281 1.066406 C 101.546875 1.066406 100.28125 2.335938 100.28125 3.902344 C 100.28125 5.46875 101.546875 6.738281 103.113281 6.738281 C 104.679688 6.738281 105.949219 5.46875 105.949219 3.902344 Z M 105.949219 3.902344 "></path> <g clip-path="url(#H23pwce3oeQ5znPcs93QPCghKbw=-clip-0)"> <path fill="none" stroke-width="0.3985" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 17.008219 24.548656 C 17.008219 26.115062 15.738687 27.384594 14.172281 27.384594 C 12.605875 27.384594 11.34025 26.115062 11.34025 24.548656 C 11.34025 22.98225 12.605875 21.712719 14.172281 21.712719 C 15.738687 21.712719 17.008219 22.98225 17.008219 24.548656 Z M 17.008219 24.548656 " transform="matrix(1, 0, 0, -1, 88.941, 28.451)"></path> </g> <path fill-rule="nonzero" fill="rgb(0%, 0%, 0%)" fill-opacity="1" d="M 105.949219 53 C 105.949219 51.433594 104.679688 50.164062 103.113281 50.164062 C 101.546875 50.164062 100.28125 51.433594 100.28125 53 C 100.28125 54.566406 101.546875 55.835938 103.113281 55.835938 C 104.679688 55.835938 105.949219 54.566406 105.949219 53 Z M 105.949219 53 "></path> <g clip-path="url(#H23pwce3oeQ5znPcs93QPCghKbw=-clip-1)"> <path fill="none" stroke-width="0.3985" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 17.008219 -24.549 C 17.008219 -22.982594 15.738687 -21.713063 14.172281 -21.713063 C 12.605875 -21.713063 11.34025 -22.982594 11.34025 -24.549 C 11.34025 -26.115406 12.605875 -27.384938 14.172281 -27.384938 C 15.738687 -27.384938 17.008219 -26.115406 17.008219 -24.549 Z M 17.008219 -24.549 " transform="matrix(1, 0, 0, -1, 88.941, 28.451)"></path> </g> <path fill-rule="nonzero" fill="rgb(0%, 0%, 0%)" fill-opacity="1" stroke-width="0.3985" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -25.511313 0.00178125 C -25.511313 1.564281 -26.780844 2.833812 -28.34725 2.833812 C -29.913656 2.833812 -31.183188 1.564281 -31.183188 0.00178125 C -31.183188 -1.564625 -29.913656 -2.834156 -28.34725 -2.834156 C -26.780844 -2.834156 -25.511313 -1.564625 -25.511313 0.00178125 Z M -25.511313 0.00178125 " transform="matrix(1, 0, 0, -1, 88.941, 28.451)"></path> <path fill-rule="nonzero" fill="rgb(0%, 0%, 0%)" fill-opacity="1" stroke-width="0.3985" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -53.858969 0.00178125 C -53.858969 1.564281 -55.1285 2.833812 -56.694906 2.833812 C -58.257406 2.833812 -59.526938 1.564281 -59.526938 0.00178125 C -59.526938 -1.564625 -58.257406 -2.834156 -56.694906 -2.834156 C -55.1285 -2.834156 -53.858969 -1.564625 -53.858969 0.00178125 Z M -53.858969 0.00178125 " transform="matrix(1, 0, 0, -1, 88.941, 28.451)"></path> <path fill-rule="nonzero" fill="rgb(0%, 0%, 0%)" fill-opacity="1" stroke-width="0.3985" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -82.206625 0.00178125 C -82.206625 1.564281 -83.476156 2.833812 -85.038656 2.833812 C -86.605063 2.833812 -87.874594 1.564281 -87.874594 0.00178125 C -87.874594 -1.564625 -86.605063 -2.834156 -85.038656 -2.834156 C -83.476156 -2.834156 -82.206625 -1.564625 -82.206625 0.00178125 Z M -82.206625 0.00178125 " transform="matrix(1, 0, 0, -1, 88.941, 28.451)"></path> <path fill="none" stroke-width="0.3985" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 2.367594 4.099437 L 11.805094 20.447094 " transform="matrix(1, 0, 0, -1, 88.941, 28.451)"></path> <path fill="none" stroke-width="0.3985" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 2.367594 -4.099781 L 11.805094 -20.447438 " transform="matrix(1, 0, 0, -1, 88.941, 28.451)"></path> <path fill="none" stroke-width="0.3985" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -4.101156 0.00178125 L -24.245688 0.00178125 " transform="matrix(1, 0, 0, -1, 88.941, 28.451)"></path> <path fill="none" stroke-width="0.3985" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -52.593344 0.00178125 L -32.448813 0.00178125 " transform="matrix(1, 0, 0, -1, 88.941, 28.451)"></path> <path fill="none" stroke-width="0.3985" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -80.941 0.00178125 L -60.792563 0.00178125 " transform="matrix(1, 0, 0, -1, 88.941, 28.451)"></path> </svg> <h2 id="related_concepts">Related concepts</h2> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/ADE+classification">ADE classification</a></strong> and <strong><a class="existingWikiWord" href="/nlab/show/McKay+correspondence">McKay correspondence</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/Dynkin+diagram">Dynkin diagram</a>/ <br /> <a class="existingWikiWord" href="/nlab/show/Dynkin+quiver">Dynkin quiver</a></th><th><a class="existingWikiWord" href="/nlab/show/dihedron">dihedron</a>,<br /> <a class="existingWikiWord" href="/nlab/show/Platonic+solid">Platonic solid</a></th><th><a class="existingWikiWord" href="/nlab/show/classification+of+finite+rotation+groups">finite subgroups of SO(3)</a></th><th><a class="existingWikiWord" href="/nlab/show/classification+of+finite+rotation+groups">finite subgroups of SU(2)</a></th><th><a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">A_{n \geq 1}</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{n+1}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{n+1}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(n+1)</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/A1">A1</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cyclic+group+of+order+2">cyclic group of order 2</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cyclic+group+of+order+2">cyclic group of order 2</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SU%282%29">SU(2)</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/A2">A2</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cyclic+group+of+order+3">cyclic group of order 3</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_3</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cyclic+group+of+order+3">cyclic group of order 3</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_3</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SU%283%29">SU(3)</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/A3">A3</a> <br /> = <br /> <a class="existingWikiWord" href="/nlab/show/D3">D3</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cyclic+group+of+order+4">cyclic group of order 4</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_4</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cyclic+group+of+order+4">cyclic group of order 4</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><msub><mi>D</mi> <mn>2</mn></msub><mo>≃</mo><msub><mi>ℤ</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">2 D_2 \simeq \mathbb{Z}_4</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SU%284%29">SU(4)</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math> <br /> <a class="existingWikiWord" href="/nlab/show/Spin%286%29">Spin(6)</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D4">D4</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dihedron">dihedron</a> on <br /> <a class="existingWikiWord" href="/nlab/show/bigon">bigon</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Klein+four-group">Klein four-group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mn>4</mn></msub><mo>≃</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>×</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quaternion+group">quaternion group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><msub><mi>D</mi> <mn>4</mn></msub><mo>≃</mo></mrow><annotation encoding="application/x-tex">2 D_4 \simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Q8">Q8</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%288%29">SO(8)</a>, <a class="existingWikiWord" href="/nlab/show/Spin%288%29">Spin(8)</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D5">D5</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dihedron">dihedron</a> on <br /> <a class="existingWikiWord" href="/nlab/show/triangle">triangle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dihedral+group+of+order+6">dihedral group of order 6</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mn>6</mn></msub></mrow><annotation encoding="application/x-tex">D_6</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/binary+dihedral+group+of+order+12">binary dihedral group of order 12</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><msub><mi>D</mi> <mn>6</mn></msub></mrow><annotation encoding="application/x-tex">2 D_6</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2810%29">SO(10)</a>, <a class="existingWikiWord" href="/nlab/show/Spin%2810%29">Spin(10)</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D6">D6</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dihedron">dihedron</a> on <br /> <a class="existingWikiWord" href="/nlab/show/square">square</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dihedral+group+of+order+8">dihedral group of order 8</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">D_8</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/binary+dihedral+group+of+order+16">binary dihedral group of order 16</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><msub><mi>D</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">2 D_{8}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2812%29">SO(12)</a>, <a class="existingWikiWord" href="/nlab/show/Spin%2812%29">Spin(12)</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mrow><mi>n</mi><mo>≥</mo><mn>4</mn></mrow></msub></mrow><annotation encoding="application/x-tex">D_{n \geq 4}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dihedron">dihedron</a>, <br /> <a class="existingWikiWord" href="/nlab/show/hosohedron">hosohedron</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dihedral+group">dihedral group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mrow><mn>2</mn><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">D_{2(n-2)}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/binary+dihedral+group">binary dihedral group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><msub><mi>D</mi> <mrow><mn>2</mn><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">2 D_{2(n-2)}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a>, <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(2n)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(2n)</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>6</mn></msub></mrow><annotation encoding="application/x-tex">E_6</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tetrahedron">tetrahedron</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tetrahedral+group">tetrahedral group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/binary+tetrahedral+group">binary tetrahedral group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>T</mi></mrow><annotation encoding="application/x-tex">2T</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/E6">E6</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>7</mn></msub></mrow><annotation encoding="application/x-tex">E_7</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cube">cube</a>, <br /> <a class="existingWikiWord" href="/nlab/show/octahedron">octahedron</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/octahedral+group">octahedral group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi></mrow><annotation encoding="application/x-tex">O</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/binary+octahedral+group">binary octahedral group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>O</mi></mrow><annotation encoding="application/x-tex">2O</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/E7">E7</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dodecahedron">dodecahedron</a>, <br /> <a class="existingWikiWord" href="/nlab/show/icosahedron">icosahedron</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/icosahedral+group">icosahedral group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/binary+icosahedral+group">binary icosahedral group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>I</mi></mrow><annotation encoding="application/x-tex">2I</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/E8">E8</a></td></tr> </tbody></table> </div></body></html> </div> <div class="revisedby"> <p> Created on August 29, 2019 at 08:22:36. See the <a href="/nlab/history/D6" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/D6" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/10273/#Item_1">Discuss</a> <a href="/nlab/show/D6/cite" style="color: black">Cite</a> <a href="/nlab/print/D6" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/D6" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>