Impressum chap3.html
Sprache: HTML
|
|
| products/sources/formale Sprachen/GAP/pkg/atlasrep/doc/chap3.html |
 |
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<title>GAP (AtlasRep) - Chapter 3: The User Interface of the AtlasRep Package</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chap3" onload="jscontent()">
<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chap5.html">5</a> <a href="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div>
<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap2.html">[Previous Chapter]</a> <a href="chap4.html">[Next Chapter]</a> </div>
<p id="mathjaxlink" class="pcenter"><a href="chap3_mj.html">[MathJax on]</a></p>
<p><a id="X87EAF8E578D95793" name="X87EAF8E578D95793"></a></p>
<div class="ChapSects"><a href="chap3.html#X87EAF8E578D95793">3 <span class="Heading">The User Interface of the <strong class="pkg">AtlasRep</strong> Package</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X87D26B13819A8209">3.1 <span class="Heading">Accessing vs. Constructing Representations</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X81BF52FC7B8C08D4">3.2 <span class="Heading">Group Names Used in the <strong class="pkg">AtlasRep</strong> Package</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X795DB7E486E0817D">3.3 <span class="Heading">Standard Generators Used in the <strong class="pkg">AtlasRep</strong> Package</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X861CD545803B97E8">3.4 <span class="Heading">Class Names Used in the <strong class="pkg">AtlasRep</strong> Package</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X850EEDEE831EE039">3.4-1 <span class="Heading">Definition of <strong class="pkg">ATLAS</strong> Class Names</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X78166D1D7D18EFBF">3.4-2 AtlasClassNames</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7B14A254870BA5A1">3.4-3 AtlasCharacterNames</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7CC88B2287A72204">3.5 <span class="Heading">Accessing Data via <strong class="pkg">AtlasRep</strong></span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X79DACFFA7E2D1A99">3.5-1 DisplayAtlasInfo</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7CE4FF2380DB47F2">3.5-2 <span class="Heading">Examples for DisplayAtlasInfo</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7D1CCCF8852DFF39">3.5-3 AtlasGenerators</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X801F2E657C8A79ED">3.5-4 AtlasProgram</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X83DFD8967E6BC831">3.5-5 AtlasProgramInfo</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X841478AB7CD06D44">3.5-6 OneAtlasGeneratingSetInfo</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X84C2D76482E60E42">3.5-7 AllAtlasGeneratingSetInfos</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X80AABEE783363B70">3.5-8 <span class="Heading">AtlasGroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7A3E460C82B3D9A3">3.5-9 <span class="Heading">AtlasSubgroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X87BC7D9C7BA2F27A">3.5-10 AtlasRepInfoRecord</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X87B012B080D01413">3.5-11 <span class="Heading">EvaluatePresentation</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X79F63403821C1E24">3.5-12 <span class="Heading">StandardGeneratorsData</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X790D5F8C7E8E6947">3.6 <span class="Heading"><strong class="pkg">Browse</strong> Applications Provided by <strong class="pkg">AtlasRep</strong></span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7F31A7CB841FE63F">3.6-1 BrowseMinimalDegrees</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X84ED4FC182C28198">3.6-2 BrowseBibliographySporadicSimple</a></span>
</div></div>
</div>
<h3>3 <span class="Heading">The User Interface of the <strong class="pkg">AtlasRep</strong> Package</span></h3>
<p>The <em>user interface</em> is the part of the <strong class="pkg">GAP</strong> interface that allows one to display information about the current contents of the database and to access individual data (perhaps by downloading them, see Section <a href="chap4.html#X7C3293A98577EE68"><span class="RefLink">4.2-1</span></a>). The corresponding functions are described in this chapter. See Section <a href="chap2.html#X87ACE06E82B68589"><span class="RefLink">2.4</span></a> for some small examples how to use the functions of the interface.</p>
<p>Data extensions of the <strong class="pkg">AtlasRep</strong> package are regarded as another part of the <strong class="pkg">GAP</strong> interface, they are described in Chapter <a href="chap5.html#X7B0718A178BB10CA"><span class="RefLink">5</span></a>. Finally, the low level part of the interface is described in Chapter <a href="chap7.html#X7F77634D817156B3"><span class="RefLink">7</span></a>.</p>
<p><a id="X87D26B13819A8209" name="X87D26B13819A8209"></a></p>
<h4>3.1 <span class="Heading">Accessing vs. Constructing Representations</span></h4>
<p>Note that <em>accessing</em> the data means in particular that it is <em>not</em> the aim of this package to <em>construct</em> representations from known ones. For example, if at least one permutation representation for a group <span class="SimpleMath">G</span> is stored but no matrix representation in a positive characteristic <span class="SimpleMath">p</span>, say, then <code class="func">OneAtlasGeneratingSetInfo</code> (<a href="chap3.html#X841478AB7CD06D44"><span class="RefLink">3.5-6</span></a>) returns <code class="keyw">fail</code> when it is asked for a description of an available set of matrix generators for <span class="SimpleMath">G</span> in characteristic <span class="SimpleMath">p</span>, although such a representation can be obtained by reduction modulo <span class="SimpleMath">p</span> of an integral matrix representation, which in turn can be constructed from any permutation representation.</p>
<p><a id="X81BF52FC7B8C08D4" name="X81BF52FC7B8C08D4"></a></p>
<h4>3.2 <span class="Heading">Group Names Used in the <strong class="pkg">AtlasRep</strong> Package</span></h4>
<p><a id="sect:groupnames"/> When you access data via the <strong class="pkg">AtlasRep</strong> package, you specify the group in question by an admissible <em>name</em>. Thus it is essential to know these names, which are called <em>the <strong class="pkg">GAP</strong> names</em> of the group in the following.</p>
<p>For a group <span class="SimpleMath">G</span>, say, whose character table is available in <strong class="pkg">GAP</strong>'s Character Table Library (see [Bre22]), the admissible names of G are the admissible names of this character table. One such name is the Identifier (Reference: Identifier for character tables) value of the character table, see CTblLib: Admissible Names for Character Tables in CTblLib. This name is usually very similar to the name used in the ATLAS of Finite Groups [CCN+85]. For example, "M22" is a GAP name of the Mathieu group M_22, "12_1.U4(3).2_1" is a GAP name of 12_1.U_4(3).2_1, the two names "S5" and "A5.2" are GAP names of the symmetric group S_5, and the two names "F3+" and "Fi24'" are GAP names of the simple Fischer group Fi_24^'.
<p>When a <strong class="pkg">GAP</strong> name is required as an input of a package function, this input is case insensitive. For example, both <code class="code">"A5"</code> and <code class="code">"a5"</code> are valid arguments of <code class="func">DisplayAtlasInfo</code> (<a href="chap3.html#X79DACFFA7E2D1A99"><span class="RefLink">3.5-1</span></a>).</p>
<p>Internally, for example as part of filenames (see Section <a href="chap7.html#X7A86627B80980F61"><span class="RefLink">7.6</span></a>), the package uses names that may differ from the <strong class="pkg">GAP</strong> names; these names are called <em><strong class="pkg">ATLAS</strong>-file names</em>. For example, <code class="code">"A5"</code>, <code class="code">"TE62"</code>, and <code class="code">"F24"</code> are <strong class="pkg">ATLAS</strong>-file names. Of these, only <code class="code">"A5"</code> is also a <strong class="pkg">GAP</strong> name, but the other two are not; corresponding <strong class="pkg">GAP</strong> names are <code class="code">"2E6(2)"</code> and <code class="code">"Fi24'"</code>, respectively.</p>
<p><a id="X795DB7E486E0817D" name="X795DB7E486E0817D"></a></p>
<h4>3.3 <span class="Heading">Standard Generators Used in the <strong class="pkg">AtlasRep</strong> Package</span></h4>
<p>For the general definition of <em>standard generators</em> of a group, see <a href="chapBib.html#biBWil96">[Wil96]</a>.</p>
<p>Several <em>different</em> standard generators may be defined for a group, the definitions for each group that occurs in the <strong class="pkg">ATLAS</strong> of Group Representations can be found at</p>
<p><span class="URL"><a href="http://atlas.math.rwth-aachen.de/Atlas/v3">http://atlas.math.rwth-aachen.de/Atlas/v3</a></span>.</p>
<p>When one specifies the standardization, the <span class="SimpleMath">i</span>-th set of standard generators is denoted by the number <span class="SimpleMath">i</span>. Note that when more than one set of standard generators is defined for a group, one must be careful to use <em>compatible standardization</em>. For example, the straight line programs, straight line decisions and black box programs in the database refer to a specific standardization of their inputs. That is, a straight line program for computing generators of a certain subgroup of a group <span class="SimpleMath">G</span> is defined only for a specific set of standard generators of <span class="SimpleMath">G</span>, and applying the program to matrix or permutation generators of <span class="SimpleMath">G</span> but w. r. t. a different standardization may yield unpredictable results. Therefore the results returned by the functions described in this chapter contain information about the standardizations they refer to.</p>
<p><a id="X861CD545803B97E8" name="X861CD545803B97E8"></a></p>
<h4>3.4 <span class="Heading">Class Names Used in the <strong class="pkg">AtlasRep</strong> Package</span></h4>
<p>For each straight line program (see <code class="func">AtlasProgram</code> (<a href="chap3.html#X801F2E657C8A79ED"><span class="RefLink">3.5-4</span></a>)) that is used to compute lists of class representatives, it is essential to describe the classes in which these elements lie. Therefore, in these cases the records returned by the function <code class="func">AtlasProgram</code> (<a href="chap3.html#X801F2E657C8A79ED"><span class="RefLink">3.5-4</span></a>) contain a component <code class="code">outputs</code> with value a list of <em>class names</em>.</p>
<p>Currently we define these class names only for simple groups and certain extensions of simple groups, see Section <a href="chap3.html#X850EEDEE831EE039"><span class="RefLink">3.4-1</span></a>. The function <code class="func">AtlasClassNames</code> (<a href="chap3.html#X78166D1D7D18EFBF"><span class="RefLink">3.4-2</span></a>) can be used to compute the list of class names from the character table in the <strong class="pkg">GAP</strong> Library.</p>
<p><a id="X850EEDEE831EE039" name="X850EEDEE831EE039"></a></p>
<h5>3.4-1 <span class="Heading">Definition of <strong class="pkg">ATLAS</strong> Class Names</span></h5>
<p>For the definition of class names of an almost simple group, we assume that the ordinary character tables of all nontrivial normal subgroups are shown in the <strong class="pkg">ATLAS</strong> of Finite Groups <a href="chapBib.html#biBCCN85">[CCN+85]</a>.</p>
<p>Each class name is a string consisting of the element order of the class in question followed by a combination of capital letters, digits, and the characters <code class="code">' and - (starting with a capital letter). For example, 1A, 12A1, and 3B'</code> denote the class that contains the identity element, a class of element order <span class="SimpleMath">12</span>, and a class of element order <span class="SimpleMath">3</span>, respectively.</p>
<ol>
<li><p>For the table of a <em>simple</em> group, the class names are the same as returned by the two argument version of the <strong class="pkg">GAP</strong> function <code class="func">ClassNames</code> (<a href="../../../doc/ref/chap71.html#X804CFD597C795801"><span class="RefLink">Reference: ClassNames</span></a>), cf. <a href="chapBib.html#biBCCN85">[CCN+85, Chapter 7, Section 5]</a>: The classes are arranged w. r. t. increasing element order and for each element order w. r. t. decreasing centralizer order, the conjugacy classes that contain elements of order <span class="SimpleMath">n</span> are named <span class="SimpleMath">n</span><code class="code">A</code>, <span class="SimpleMath">n</span><code class="code">B</code>, <span class="SimpleMath">n</span><code class="code">C</code>, <span class="SimpleMath">...</span>; the alphabet used here is potentially infinite, and reads <code class="code">A</code>, <code class="code">B</code>, <code class="code">C</code>, <span class="SimpleMath">...</span>, <code class="code">Z</code>, <code class="code">A1</code>, <code class="code">B1</code>, <span class="SimpleMath">...</span>, <code class="code">A2</code>, <code class="code">B2</code>, <span class="SimpleMath">...</span>.</p>
<p>For example, the classes of the alternating group <span class="SimpleMath">A_5</span> have the names <code class="code">1A</code>, <code class="code">2A</code>, <code class="code">3A</code>, <code class="code">5A</code>, and <code class="code">5B</code>.</p>
</li>
<li><p>Next we consider the case of an <em>upward extension</em> <span class="SimpleMath">G.A</span> of a simple group <span class="SimpleMath">G</span> by a <em>cyclic</em> group of order <span class="SimpleMath">A</span>. The <strong class="pkg">ATLAS</strong> defines class names for each element <span class="SimpleMath">g</span> of <span class="SimpleMath">G.A</span> only w. r. t. the group <span class="SimpleMath">G.a</span>, say, that is generated by <span class="SimpleMath">G</span> and <span class="SimpleMath">g</span>; namely, there is a power of <span class="SimpleMath">g</span> (with the exponent coprime to the order of <span class="SimpleMath">g</span>) for which the class has a name of the same form as the class names for simple groups, and the name of the class of <span class="SimpleMath">g</span> w. r. t. <span class="SimpleMath">G.a</span> is then obtained from this name by appending a suitable number of dashes <code class="code">'. So dashed class names refer exactly to those classes that are not printed in the ATLAS.
<p>For example, those classes of the symmetric group <span class="SimpleMath">S_5</span> that do not lie in <span class="SimpleMath">A_5</span> have the names <code class="code">2B</code>, <code class="code">4A</code>, and <code class="code">6A</code>. The outer classes of the group <span class="SimpleMath">L_2(8).3</span> have the names <code class="code">3B</code>, <code class="code">6A</code>, <code class="code">9D</code>, and <code class="code">3B', 6A'</code>, <code class="code">9D'. The outer elements of order 5 in the group Sz(32).5 lie in the classes with names 5B, 5B'</code>, <code class="code">5B''</code>, and <code class="code">5B'''.
<p>In the group <span class="SimpleMath">G.A</span>, the class of <span class="SimpleMath">g</span> may fuse with other classes. The name of the class of <span class="SimpleMath">g</span> in <span class="SimpleMath">G.A</span> is obtained from the names of the involved classes of <span class="SimpleMath">G.a</span> by concatenating their names after removing the element order part from all of them except the first one.</p>
<p>For example, the elements of order <span class="SimpleMath">9</span> in the group <span class="SimpleMath">L_2(27).6</span> are contained in the subgroup <span class="SimpleMath">L_2(27).3</span> but not in <span class="SimpleMath">L_2(27)</span>. In <span class="SimpleMath">L_2(27).3</span>, they lie in the classes <code class="code">9A</code>, <code class="code">9A', 9B, and 9B'</code>; in <span class="SimpleMath">L_2(27).6</span>, these classes fuse to <code class="code">9AB</code> and <code class="code">9A'B'</code>.</p>
</li>
<li><p>Now we define class names for <em>general upward extensions</em> <span class="SimpleMath">G.A</span> of a simple group <span class="SimpleMath">G</span>. Each element <span class="SimpleMath">g</span> of such a group lies in an upward extension <span class="SimpleMath">G.a</span> by a cyclic group, and the class names w. r. t. <span class="SimpleMath">G.a</span> are already defined. The name of the class of <span class="SimpleMath">g</span> in <span class="SimpleMath">G.A</span> is obtained by concatenating the names of the classes in the orbit of <span class="SimpleMath">G.A</span> on the classes of cyclic upward extensions of <span class="SimpleMath">G</span>, after ordering the names lexicographically and removing the element order part from all of them except the first one. An <em>exception</em> is the situation where dashed and non-dashed class names appear in an orbit; in this case, the dashed names are omitted.</p>
<p>For example, the classes <code class="code">21A</code> and <code class="code">21B</code> of the group <span class="SimpleMath">U_3(5).3</span> fuse in <span class="SimpleMath">U_3(5).S_3</span> to the class <code class="code">21AB</code>, and the class <code class="code">2B</code> of <span class="SimpleMath">U_3(5).2</span> fuses with the involution classes <code class="code">2B', 2B'' in the groups U_3(5).2^'</span> and <span class="SimpleMath">U_3(5).2^{''}</span> to the class <code class="code">2B</code> of <span class="SimpleMath">U_3(5).S_3</span>.</p>
<p>It may happen that some names in the <code class="code">outputs</code> component of a record returned by <code class="func">AtlasProgram</code> (<a href="chap3.html#X801F2E657C8A79ED"><span class="RefLink">3.5-4</span></a>) do not uniquely determine the classes of the corresponding elements. For example, the (algebraically conjugate) classes <code class="code">39A</code> and <code class="code">39B</code> of the group <span class="SimpleMath">Co_1</span> have not been distinguished yet. In such cases, the names used contain a minus sign <code class="code">-</code>, and mean <q>one of the classes in the range described by the name before and the name after the minus sign</q>; the element order part of the name does not appear after the minus sign. So the name <code class="code">39A-B</code> for the group <span class="SimpleMath">Co_1</span> means <code class="code">39A</code> or <code class="code">39B</code>, and the name <code class="code">20A-B''' for the group Sz(32).5 means one of the classes of element order 20 in this group (these classes lie outside the simple group Sz).
</li>
<li><p>For a <em>downward extension</em> <span class="SimpleMath">m.G.A</span> of an almost simple group <span class="SimpleMath">G.A</span> by a cyclic group of order <span class="SimpleMath">m</span>, let <span class="SimpleMath">π</span> denote the natural epimorphism from <span class="SimpleMath">m.G.A</span> onto <span class="SimpleMath">G.A</span>. Each class name of <span class="SimpleMath">m.G.A</span> has the form <code class="code">nX_0</code>, <code class="code">nX_1</code> etc., where <code class="code">nX</code> is the class name of the image under <span class="SimpleMath">π</span>, and the indices <code class="code">0</code>, <code class="code">1</code> etc. are chosen according to the position of the class in the lifting order rows for <span class="SimpleMath">G</span>, see <a href="chapBib.html#biBCCN85">[CCN+85, Chapter 7, Section 7, and the example in Section 8]</a>).</p>
<p>For example, if <span class="SimpleMath">m = 6</span> then <code class="code">1A_1</code> and <code class="code">1A_5</code> denote the classes containing the generators of the kernel of <span class="SimpleMath">π</span>, that is, central elements of order <span class="SimpleMath">6</span>.</p>
</li>
</ol>
<p><a id="X78166D1D7D18EFBF" name="X78166D1D7D18EFBF"></a></p>
<h5>3.4-2 AtlasClassNames</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AtlasClassNames</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a list of class names.</p>
<p>Let <var class="Arg">tbl</var> be the ordinary or modular character table of a group <span class="SimpleMath">G</span>, say, that is almost simple or a downward extension of an almost simple group and such that <var class="Arg">tbl</var> is an <strong class="pkg">ATLAS</strong> table from the <strong class="pkg">GAP</strong> Character Table Library, according to its <code class="func">InfoText</code> (<a href="../../../doc/ref/chap12.html#X871562FD7F982C12"><span class="RefLink">Reference: InfoText</span></a>) value. Then <code class="func">AtlasClassNames</code> returns the list of class names for <span class="SimpleMath">G</span>, as defined in Section <a href="chap3.html#X850EEDEE831EE039"><span class="RefLink">3.4-1</span></a>. The ordering of class names is the same as the ordering of the columns of <var class="Arg">tbl</var>.</p>
<p>(The function may work also for character tables that are not <strong class="pkg">ATLAS</strong> tables, but then clearly the class names returned are somewhat arbitrary.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasClassNames( CharacterTable( "L3(4).3" ) );</span>
[ "1A", "2A", "3A", "4ABC", "5A", "5B", "7A", "7B", "3B", "3B'",
"3C", "3C'", "6B", "6B'", "15A", "15A'", "15B", "15B'", "21A",
"21A'", "21B", "21B'" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasClassNames( CharacterTable( "U3(5).2" ) );</span>
[ "1A", "2A", "3A", "4A", "5A", "5B", "5CD", "6A", "7AB", "8AB",
"10A", "2B", "4B", "6D", "8C", "10B", "12B", "20A", "20B" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasClassNames( CharacterTable( "L2(27).6" ) );</span>
[ "1A", "2A", "3AB", "7ABC", "13ABC", "13DEF", "14ABC", "2B", "4A",
"26ABC", "26DEF", "28ABC", "28DEF", "3C", "3C'", "6A", "6A'",
"9AB", "9A'B'", "6B", "6B'", "12A", "12A'" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasClassNames( CharacterTable( "L3(4).3.2_2" ) );</span>
[ "1A", "2A", "3A", "4ABC", "5AB", "7A", "7B", "3B", "3C", "6B",
"15A", "15B", "21A", "21B", "2C", "4E", "6E", "8D", "14A", "14B" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasClassNames( CharacterTable( "3.A6" ) );</span>
[ "1A_0", "1A_1", "1A_2", "2A_0", "2A_1", "2A_2", "3A_0", "3B_0",
"4A_0", "4A_1", "4A_2", "5A_0", "5A_1", "5A_2", "5B_0", "5B_1",
"5B_2" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasClassNames( CharacterTable( "2.A5.2" ) );</span>
[ "1A_0", "1A_1", "2A_0", "3A_0", "3A_1", "5AB_0", "5AB_1", "2B_0",
"4A_0", "4A_1", "6A_0", "6A_1" ]
</pre></div>
<p><a id="X7B14A254870BA5A1" name="X7B14A254870BA5A1"></a></p>
<h5>3.4-3 AtlasCharacterNames</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AtlasCharacterNames</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a list of character names.</p>
<p>Let <var class="Arg">tbl</var> be the ordinary or modular character table of a simple group. <code class="func">AtlasCharacterNames</code> returns a list of strings, the <span class="SimpleMath">i</span>-th entry being the name of the <span class="SimpleMath">i</span>-th irreducible character of <var class="Arg">tbl</var>; this name consists of the degree of this character followed by distinguishing lowercase letters.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasCharacterNames( CharacterTable( "A5" ) ); </span>
[ "1a", "3a", "3b", "4a", "5a" ]
</pre></div>
<p><a id="X7CC88B2287A72204" name="X7CC88B2287A72204"></a></p>
<h4>3.5 <span class="Heading">Accessing Data via <strong class="pkg">AtlasRep</strong></span></h4>
<p>The examples shown in this section refer to the situation that no extensions have been notified, and to a perhaps outdated table of contents. That is, the current version of the database may contain more information than is shown here.</p>
<p><a id="X79DACFFA7E2D1A99" name="X79DACFFA7E2D1A99"></a></p>
<h5>3.5-1 DisplayAtlasInfo</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DisplayAtlasInfo</code>( [<var class="Arg">listofnames</var>][,] [<var class="Arg">std</var>][,] [<var class="Arg">"contents"</var>, <var class="Arg">sources</var>][,] [<var class="Arg">...</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DisplayAtlasInfo</code>( <var class="Arg">gapname</var>[, <var class="Arg">std</var>][, <var class="Arg">...</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function lists the information available via the <strong class="pkg">AtlasRep</strong> package, for the given input.</p>
<p>There are essentially three ways of calling this function.</p>
<ul>
<li><p>If there is no argument or if the first argument is a list <var class="Arg">listofnames</var> of strings that are <strong class="pkg">GAP</strong> names of groups, <code class="func">DisplayAtlasInfo</code> shows an overview of the known information.</p>
</li>
<li><p>If the first argument is a string <var class="Arg">gapname</var> that is a <strong class="pkg">GAP</strong> name of a group, <code class="func">DisplayAtlasInfo</code> shows an overview of the information that is available for this group.</p>
</li>
<li><p>If the string <code class="code">"contents"</code> is the only argument then the function shows which parts of the database are available; these are at least the <code class="code">"core"</code> part, which means the data from the <strong class="pkg">ATLAS</strong> of Group Representations, and the <code class="code">"internal"</code> part, which means the data that are distributed with the <strong class="pkg">AtlasRep</strong> package. Other parts can become available by calls to <code class="func">AtlasOfGroupRepresentationsNotifyData</code> (<a href="chap5.html#X81B5FA0578257653"><span class="RefLink">5.1-1</span></a>). Note that the shown numbers of locally available files depend on what has already been downloaded.</p>
</li>
</ul>
<p>In each case, the information will be printed to the screen or will be fed into a pager, see Section <a href="chap4.html#X81F055037F9D3068"><span class="RefLink">4.2-11</span></a>. An interactive alternative to <code class="func">DisplayAtlasInfo</code> is the function <code class="func">BrowseAtlasInfo</code> (<a href="../../../pkg/browse/doc/chap6.html#X8411AF928194C5AB"><span class="RefLink">Browse: BrowseAtlasInfo</span></a>), see <a href="chapBib.html#biBBrowse">[BL18]</a>.</p>
<p>The following paragraphs describe the structure of the output in the two cases. Examples can be found in Section <a href="chap3.html#X7CE4FF2380DB47F2"><span class="RefLink">3.5-2</span></a>.</p>
<p>Called without arguments, <code class="func">DisplayAtlasInfo</code> shows a general overview for all groups. If some information is available for the group <span class="SimpleMath">G</span>, say, then one line is shown for <span class="SimpleMath">G</span>, with the following columns.</p>
<dl>
<dt><strong class="Mark"><code class="code">group</code></strong></dt>
<dd><p>the <strong class="pkg">GAP</strong> name of <span class="SimpleMath">G</span> (see Section <a href="chap3.html#X81BF52FC7B8C08D4"><span class="RefLink">3.2</span></a>), if applicable followed by a string (by default a star <code class="code">*</code>) indicating that at least one column refers to data not belonging to the core part of the database (see Section <a href="chap4.html#X862C660878D422FA"><span class="RefLink">4.2-12</span></a>).</p>
</dd>
<dt><strong class="Mark"><code class="code">#</code></strong></dt>
<dd><p>the number of faithful representations stored for <span class="SimpleMath">G</span> that satisfy the additional conditions given (see below),</p>
</dd>
<dt><strong class="Mark"><code class="code">maxes</code></strong></dt>
<dd><p>the number of available straight line programs for computing generators of maximal subgroups of <span class="SimpleMath">G</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">cl</code></strong></dt>
<dd><p>a <code class="code">+</code> sign if at least one program for computing representatives of conjugacy classes of elements of <span class="SimpleMath">G</span> is stored,</p>
</dd>
<dt><strong class="Mark"><code class="code">cyc</code></strong></dt>
<dd><p>a <code class="code">+</code> sign if at least one program for computing representatives of classes of maximally cyclic subgroups of <span class="SimpleMath">G</span> is stored,</p>
</dd>
<dt><strong class="Mark"><code class="code">out</code></strong></dt>
<dd><p>descriptions of outer automorphisms of <span class="SimpleMath">G</span> for which at least one program is stored,</p>
</dd>
<dt><strong class="Mark"><code class="code">fnd</code></strong></dt>
<dd><p>a <code class="code">+</code> sign if at least one program is available for finding standard generators,</p>
</dd>
<dt><strong class="Mark"><code class="code">chk</code></strong></dt>
<dd><p>a <code class="code">+</code> sign if at least one program is available for checking whether a set of generators is a set of standard generators, and</p>
</dd>
<dt><strong class="Mark"><code class="code">prs</code></strong></dt>
<dd><p>a <code class="code">+</code> sign if at least one program is available that encodes a presentation.</p>
</dd>
</dl>
<p>Called with a list <var class="Arg">listofnames</var> of strings that are <strong class="pkg">GAP</strong> names of some groups, <code class="func">DisplayAtlasInfo</code> prints the overview described above but restricted to the groups in this list.</p>
<p>In addition to or instead of <var class="Arg">listofnames</var>, the string <code class="code">"contents"</code> and a description <span class="SimpleMath">sources</span> of the data may be given about which the overview is formed. See below for admissible values of <span class="SimpleMath">sources</span>.</p>
<p>Called with a string <var class="Arg">gapname</var> that is a <strong class="pkg">GAP</strong> name of a group, <code class="func">DisplayAtlasInfo</code> prints an overview of the information that is available for this group. One line is printed for each faithful representation, showing the number of this representation (which can be used in calls of <code class="func">AtlasGenerators</code> (<a href="chap3.html#X7D1CCCF8852DFF39"><span class="RefLink">3.5-3</span></a>)), and a string of one of the following forms; in both cases, <span class="SimpleMath">id</span> is a (possibly empty) string.</p>
<dl>
<dt><strong class="Mark"><code class="code">G <= Sym(</code><span class="SimpleMath">n</span><span class="SimpleMath">id</span><code class="code">)</code></strong></dt>
<dd><p>denotes a permutation representation of degree <span class="SimpleMath">n</span>, for example <code class="code">G <= Sym(40a)</code> and <code class="code">G <= Sym(40b)</code> denote two (nonequivalent) representations of degree <span class="SimpleMath">40</span>.</p>
</dd>
<dt><strong class="Mark"><code class="code">G <= GL(</code><span class="SimpleMath">n</span><span class="SimpleMath">id</span>,<span class="SimpleMath">descr</span><code class="code">)</code></strong></dt>
<dd><p>denotes a matrix representation of dimension <span class="SimpleMath">n</span> over a coefficient ring described by <span class="SimpleMath">descr</span>, which can be a prime power, <code class="code">ℤ</code> (denoting the ring of integers), a description of an algebraic extension field, <code class="code">ℂ</code> (denoting an unspecified algebraic extension field), or <code class="code">ℤ/</code><span class="SimpleMath">m</span><code class="code">ℤ</code> for an integer <span class="SimpleMath">m</span> (denoting the ring of residues mod <span class="SimpleMath">m</span>); for example, <code class="code">G <= GL(2a,4)</code> and <code class="code">G <= GL(2b,4)</code> denote two (nonequivalent) representations of dimension <span class="SimpleMath">2</span> over the field with four elements.</p>
</dd>
</dl>
<p>Below the representations, the programs available for <var class="Arg">gapname</var> are listed. In each row of the overview, the entry in the first column is followed by a string (by default a star <code class="code">*</code>) if the row refers to data not belonging to the core part of the database (see Section <a href="chap4.html#X862C660878D422FA"><span class="RefLink">4.2-12</span></a>).</p>
<p>The following optional arguments can be used to restrict the overviews.</p>
<dl>
<dt><strong class="Mark"><var class="Arg">std</var></strong></dt>
<dd><p>must be a positive integer or a list of positive integers; if it is given then only those representations are considered that refer to the <var class="Arg">std</var>-th set of standard generators or the <span class="SimpleMath">i</span>-th set of standard generators, for <span class="SimpleMath">i</span> in <var class="Arg">std</var> (see Section <a href="chap3.html#X795DB7E486E0817D"><span class="RefLink">3.3</span></a>),</p>
</dd>
<dt><strong class="Mark"><code class="code">"contents"</code> and <span class="SimpleMath">sources</span></strong></dt>
<dd><p>for a string or a list of strings <span class="SimpleMath">sources</span>, restrict the data about which the overview is formed; if <span class="SimpleMath">sources</span> is the string <code class="code">"core"</code> then only data from the <strong class="pkg">ATLAS</strong> of Group Representations are considered, if <span class="SimpleMath">sources</span> is a string that denotes a data extension in the sense of a <code class="code">dirid</code> argument of <code class="func">AtlasOfGroupRepresentationsNotifyData</code> (<a href="chap5.html#X81B5FA0578257653"><span class="RefLink">5.1-1</span></a>) then only the data that belong to this data extension are considered; also a list of such strings may be given, then the union of these data is considered,</p>
</dd>
<dt><strong class="Mark"><code class="code">Identifier</code> and <span class="SimpleMath">id</span></strong></dt>
<dd><p>restrict to representations with <code class="code">id</code> component in the list <span class="SimpleMath">id</span> (note that this component is itself a list, entering this list is not admissible), or satisfying the function <span class="SimpleMath">id</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">IsPermGroup</code> and <code class="keyw">true</code> (or <code class="keyw">false</code>)</strong></dt>
<dd><p>restrict to permutation representations (or to representations that are not permutation representations),</p>
</dd>
<dt><strong class="Mark"><code class="code">NrMovedPoints</code> and <span class="SimpleMath">n</span></strong></dt>
<dd><p>for a positive integer, a list of positive integers, or a property <span class="SimpleMath">n</span>, restrict to permutation representations of degree equal to <span class="SimpleMath">n</span>, or in the list <span class="SimpleMath">n</span>, or satisfying the function <span class="SimpleMath">n</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">NrMovedPoints</code> and the string <code class="code">"minimal"</code></strong></dt>
<dd><p>restrict to faithful permutation representations of minimal degree (if this information is available),</p>
</dd>
<dt><strong class="Mark"><code class="code">IsTransitive</code> and a boolean value</strong></dt>
<dd><p>restrict to transitive or intransitive permutation representations where this information is available (if the value <code class="keyw">true</code> or <code class="keyw">false</code> is given), or to representations for which this information is not available (if the value <code class="keyw">fail</code> is given),</p>
</dd>
<dt><strong class="Mark"><code class="code">IsPrimitive</code> and a boolean value</strong></dt>
<dd><p>restrict to primitive or imprimitive permutation representations where this information is available (if the value <code class="keyw">true</code> or <code class="keyw">false</code> is given), or to representations for which this information is not available (if the value <code class="keyw">fail</code> is given),</p>
</dd>
<dt><strong class="Mark"><code class="code">Transitivity</code> and <span class="SimpleMath">n</span></strong></dt>
<dd><p>for a nonnegative integer, a list of nonnegative integers, or a property <span class="SimpleMath">n</span>, restrict to permutation representations for which the information is available that the transitivity is equal to <span class="SimpleMath">n</span>, or is in the list <span class="SimpleMath">n</span>, or satisfies the function <span class="SimpleMath">n</span>; if <span class="SimpleMath">n</span> is <code class="keyw">fail</code> then restrict to all permutation representations for which this information is not available,</p>
</dd>
<dt><strong class="Mark"><code class="code">RankAction</code> and <span class="SimpleMath">n</span></strong></dt>
<dd><p>for a nonnegative integer, a list of nonnegative integers, or a property <span class="SimpleMath">n</span>, restrict to permutation representations for which the information is available that the rank is equal to <span class="SimpleMath">n</span>, or is in the list <span class="SimpleMath">n</span>, or satisfies the function <span class="SimpleMath">n</span>; if <span class="SimpleMath">n</span> is <code class="keyw">fail</code> then restrict to all permutation representations for which this information is not available,</p>
</dd>
<dt><strong class="Mark"><code class="code">IsMatrixGroup</code> and <code class="keyw">true</code> (or <code class="keyw">false</code>)</strong></dt>
<dd><p>restrict to matrix representations (or to representations that are not matrix representations),</p>
</dd>
<dt><strong class="Mark"><code class="code">Characteristic</code> and <span class="SimpleMath">p</span></strong></dt>
<dd><p>for a prime integer, a list of prime integers, or a property <span class="SimpleMath">p</span>, restrict to matrix representations over fields of characteristic equal to <span class="SimpleMath">p</span>, or in the list <span class="SimpleMath">p</span>, or satisfying the function <span class="SimpleMath">p</span> (representations over residue class rings that are not fields can be addressed by entering <code class="keyw">fail</code> as the value of <span class="SimpleMath">p</span>),</p>
</dd>
<dt><strong class="Mark"><code class="code">Dimension</code> and <span class="SimpleMath">n</span></strong></dt>
<dd><p>for a positive integer, a list of positive integers, or a property <span class="SimpleMath">n</span>, restrict to matrix representations of dimension equal to <span class="SimpleMath">n</span>, or in the list <span class="SimpleMath">n</span>, or satisfying the function <span class="SimpleMath">n</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">Characteristic</code>, <span class="SimpleMath">p</span>, <code class="code">Dimension</code>,
and the string <code class="code">"minimal"</code></strong></dt>
<dd><p>for a prime integer <span class="SimpleMath">p</span>, restrict to faithful matrix representations over fields of characteristic <span class="SimpleMath">p</span> that have minimal dimension (if this information is available),</p>
</dd>
<dt><strong class="Mark"><code class="code">Ring</code> and <span class="SimpleMath">R</span></strong></dt>
<dd><p>for a ring or a property <span class="SimpleMath">R</span>, restrict to matrix representations for which the information is available that the ring spanned by the matrix entries is contained in this ring or satisfies this property (note that the representation might be defined over a proper subring); if <span class="SimpleMath">R</span> is <code class="keyw">fail</code> then restrict to all matrix representations for which this information is not available,</p>
</dd>
<dt><strong class="Mark"><code class="code">Ring</code>, <span class="SimpleMath">R</span>, <code class="code">Dimension</code>,
and the string <code class="code">"minimal"</code></strong></dt>
<dd><p>for a ring <span class="SimpleMath">R</span>, restrict to faithful matrix representations over this ring that have minimal dimension (if this information is available),</p>
</dd>
<dt><strong class="Mark"><code class="code">Character</code> and <span class="SimpleMath">chi</span></strong></dt>
<dd><p>for a class function or a list of class functions <span class="SimpleMath">chi</span>, restrict to representations with these characters (note that the underlying characteristic of the class function, see Section <a href="../../../doc/ref/chap71.html#X7F58A82F7D88000A"><span class="RefLink">Reference: UnderlyingCharacteristic</span></a>, determines the characteristic of the representation),</p>
</dd>
<dt><strong class="Mark"><code class="code">Character</code> and <span class="SimpleMath">name</span></strong></dt>
<dd><p>for a string <span class="SimpleMath">name</span>, restrict to representations for which the character is known to have this name, according to the information shown by <code class="func">DisplayAtlasInfo</code>; if the characteristic is not specified then it defaults to zero,</p>
</dd>
<dt><strong class="Mark"><code class="code">Character</code> and <span class="SimpleMath">n</span></strong></dt>
<dd><p>for a positive integer <span class="SimpleMath">n</span>, restrict to representations for which the character is known to be the <span class="SimpleMath">n</span>-th irreducible character in <strong class="pkg">GAP</strong>'s library character table of the group in question; if the characteristic is not specified then it defaults to zero,
</dd>
<dt><strong class="Mark"><code class="code">IsStraightLineProgram</code> and <code class="keyw">true</code></strong></dt>
<dd><p>restrict to straight line programs, straight line decisions (see Section <a href="chap6.html#X8121E9567A7137C9"><span class="RefLink">6.1</span></a>), and black box programs (see Section <a href="chap6.html#X7BE856BC785A9E8F"><span class="RefLink">6.2</span></a>), and</p>
</dd>
<dt><strong class="Mark"><code class="code">IsStraightLineProgram</code> and <code class="keyw">false</code></strong></dt>
<dd><p>restrict to representations.</p>
</dd>
</dl>
<p>Note that the above conditions refer only to the information that is available without accessing the representations. For example, if it is not stored in the table of contents whether a permutation representation is primitive then this representation does not match an <code class="code">IsPrimitive</code> condition in <code class="func">DisplayAtlasInfo</code>.</p>
<p>If <q>minimality</q> information is requested and no available representation matches this condition then either no minimal representation is available or the information about the minimality is missing. See <code class="func">MinimalRepresentationInfo</code> (<a href="chap6.html# | |
