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<div class="ChapSects"><a href="chap4_mj.html#X85854AF07F2F8745">4 <span class="Heading">Contents of the <strong class="pkg">GAP</strong> Character Table Library</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X80EEFF8B79856A75">4.1 <span class="Heading">Ordinary and Brauer Tables in the <strong class="pkg">GAP</strong> Character Table Library
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X8569BC8E7A9D4BCE">4.1-1 <span class="Heading">Ordinary Character Tables</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7AD048607A08C6FF">4.1-2 <span class="Heading">Brauer Tables</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X81E3F9A384365282">4.2 <span class="Heading">Generic Character Tables</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X78CD9A2D8680506B">4.2-1 <span class="Heading">Available generic character tables</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X78DA225F78F381C9">4.2-2 CharacterTableSpecialized</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7C2CB9E07990B63D">4.2-3 <span class="Heading">Components of generic character tables</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7D693E9787073E30">4.2-4 <span class="Heading">Example: The generic table of cyclic groups</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X81231BCE79486FA3">4.2-5 <span class="Heading">Example: The generic table of the general linear group GL<span class="SimpleMath">\((2,q)\)</span>
</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7F44BD4B79473085">4.3 <span class="Heading"><strong class="pkg">Atlas</strong> Tables</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7CC608CD8690F9B1">4.3-1 <span class="Heading">Improvements to the <strong class="pkg">Atlas</strong></span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7FED949A86575949">4.3-2 <span class="Heading">Power Maps</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X824823F47BB6AD6C">4.3-3 <span class="Heading">Projective Characters and Projections</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X78732CDF85FB6774">4.3-4 <span class="Heading">Tables of Isoclinic Groups</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7C4B91CD84D5CDCC">4.3-5 <span class="Heading">Ordering of Characters and Classes</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7ADC9DC980CF0685">4.3-6 AtlasLabelsOfIrreducibles</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X827044A37C04C0D1">4.3-7 <span class="Heading">Examples of the <strong class="pkg">Atlas</strong> Format for <strong class="pkg">GAP</strong> Tables</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7BC3F0B0814D5B67">4.4 <span class="Heading"><strong class="pkg">CAS</strong> Tables</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X786A80A279674E91">4.4-1 CASInfo</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X835811C279FB1E56">4.5 <span class="Heading">Customizations of the <strong class="pkg">GAP</strong> Character Table Library</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X8202ACD57ACD5CAC">4.5-1 <span class="Heading">Installing the <strong class="pkg">GAP</strong> Character Table Library</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X83FA9D6B86150501">4.5-2 <span class="Heading">Unloading Character Table Data</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7E859C3482F27089">4.5-3 <span class="Heading">Changing the display format of several functions</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7DFAD31F84F55870">4.5-4 <span class="Heading">User preference <code class="code">MagmaPath</code></span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X8782716579A1B993">4.6 <span class="Heading">Technicalities of the Access to Character Tables from the Library
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X84E18B0B84F50B1E">4.6-1 <span class="Heading">Data Files of the <strong class="pkg">GAP</strong> Character Table Library</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X84E728FD860CAC0F">4.6-2 LIBLIST</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X80B7DF9C83A0F3F1">4.6-3 LibInfoCharacterTable</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X78FFDF0F83E7EB0D">4.7 <span class="Heading">How to Extend the <strong class="pkg">GAP</strong> Character Table Library</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7A3B010A8790DD6E">4.7-1 NotifyNameOfCharacterTable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X8160EA7C85DCB485">4.7-2 LibraryFusion</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X79E06BD67F6BC3A5">4.7-3 LibraryFusionTblToTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X780CBC347876A54B">4.7-4 PrintToLib</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X79366F797CD02DAF">4.7-5 NotifyCharacterTable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X8374B5D081F85DBC">4.7-6 NotifyCharacterTables</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7E3235FD7864A672">4.8 <span class="Heading">Sanity Checks for the <strong class="pkg">GAP</strong> Character Table Library</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7D24C9D17DAB50D0">4.9 <span class="Heading">Maintenance of the <strong class="pkg">GAP</strong> Character Table Library</span></a>
</span>
</div>
</div>
<h3>4 <span class="Heading">Contents of the <strong class="pkg">GAP</strong> Character Table Library</span></h3>
<p>This chapter informs you about</p>
<ul>
<li><p>the currently available character tables (see Section <a href="chap4_mj.html#X80EEFF8B79856A75"><span class="RefLink">4.1</span></a>),</p>
</li>
<li><p>generic character tables (see Section <a href="chap4_mj.html#X81E3F9A384365282"><span class="RefLink">4.2</span></a>),</p>
</li>
<li><p>the subsets of <strong class="pkg">Atlas</strong> tables (see Section <a href="chap4_mj.html#X7F44BD4B79473085"><span class="RefLink">4.3</span></a>) and <strong class="pkg">CAS</strong> tables (see Section <a href="chap4_mj.html#X7BC3F0B0814D5B67"><span class="RefLink">4.4</span></a>),</p>
</li>
<li><p>installing the library, and related user preferences (see Section <a href="chap4_mj.html#X835811C279FB1E56"><span class="RefLink">4.5</span></a>).</p>
</li>
</ul>
<p>The following rather technical sections are thought for those who want to maintain or extend the Character Table Library.</p>
<ul>
<li><p>the technicalities of the access to library tables (see Section <a href="chap4_mj.html#X8782716579A1B993"><span class="RefLink">4.6</span></a>),</p>
</li>
<li><p>how to extend the library (see Section <a href="chap4_mj.html#X78FFDF0F83E7EB0D"><span class="RefLink">4.7</span></a>), and</p>
</li>
<li><p>sanity checks (see Section <a href="chap4_mj.html#X7E3235FD7864A672"><span class="RefLink">4.8</span></a>).</p>
</li>
</ul>
<p><a id="X80EEFF8B79856A75" name="X80EEFF8B79856A75"></a></p>
<h4>4.1 <span class="Heading">Ordinary and Brauer Tables in the <strong class="pkg">GAP</strong> Character Table Library
</span></h4>
<p>This section gives a brief overview of the contents of the <strong class="pkg">GAP</strong> character table library. For the details about, e. g., the structure of data files, see Section <a href="chap4_mj.html#X8782716579A1B993"><span class="RefLink">4.6</span></a>.</p>
<p>The changes in the character table library since the first release of <strong class="pkg">GAP</strong> 4 are listed in a file that can be fetched from</p>
<p><span class="URL"><a href="https://www.math.rwth-aachen.de/~Thomas.Breuer/ctbllib/htm/ctbldiff.htm ">https://www.math.rwth-aachen.de/~Thomas.Breuer/ctbllib/htm/ctbldiff.htm </a></span>.</p>
<p>There are three different kinds of character tables in the <strong class="pkg">GAP</strong> library, namely <em>ordinary character tables</em>, <em>Brauer tables</em>, and <em>generic character tables</em>. Note that the Brauer table and the corresponding ordinary table of a group determine the <em>decomposition matrix</em> of the group (and the decomposition matrices of its blocks). These decomposition matrices can be computed from the ordinary and modular irreducibles with <strong class="pkg">GAP</strong>, see Section <a href="../../../doc/ref/chap71_mj.html#X8733F0EA801785D4"><span class="RefLink">Reference: Operations Concerning Blocks</span></a> for details. A collection of PDF files of the known decomposition matrices of <strong class="pkg">Atlas</strong> tables in the <strong class="pkg">GAP</strong> Character Table Library can also be found at</p>
<p><span class="URL"><a href="https://www.math.rwth-aachen.de/~MOC/decomposition/">https://www.math.rwth-aachen.de/~MOC/decomposition/</a></span>.</p>
<p><a id="X8569BC8E7A9D4BCE" name="X8569BC8E7A9D4BCE"></a></p>
<h5>4.1-1 <span class="Heading">Ordinary Character Tables</span></h5>
<p>Two different aspects are useful to list the ordinary character tables available in <strong class="pkg">GAP</strong>, namely the aspect of the <em>source</em> of the tables and that of <em>relations</em> between the tables.</p>
<p>As for the source, there are first of all two big sources, namely the <strong class="pkg">Atlas</strong> of Finite Groups (see Section <a href="chap4_mj.html#X7F44BD4B79473085"><span class="RefLink">4.3</span></a>) and the <strong class="pkg">CAS</strong> library of character tables (see <a href="chapBib_mj.html#biBNPP84">[NPP84]</a>). Many <strong class="pkg">Atlas</strong> tables are contained in the <strong class="pkg">CAS</strong> library, and difficulties may arise because the succession of characters and classes in <strong class="pkg">CAS</strong> tables and <strong class="pkg">Atlas</strong> tables are in general different, so see Section <a href="chap4_mj.html#X7BC3F0B0814D5B67"><span class="RefLink">4.4</span></a> for the relations between these two variants of character tables of the same group. A subset of the <strong class="pkg">CAS</strong> tables is the set of tables of Sylow normalizers of sporadic simple groups as published in <a href="chapBib_mj.html#biBOst86">[Ost86]</a> this may be viewed as another source of character tables. The library also contains the character tables of factor groups of space groups (computed by W. Hanrath, see <a href="chapBib_mj.html#biBHan88">[Han88]</a>) that are part of <a href="chapBib_mj.html#biBHP89">[HP89]</a>, in the form of two microfiches; these tables are given in <strong class="pkg">CAS</strong> format (see Section <a href="chap4_mj.html#X7BC3F0B0814D5B67"><span class="RefLink">4.4</span></a>) on the microfiches, but they had not been part of the <q>official</q> <strong class="pkg">CAS</strong> library.</p>
<p>To avoid confusion about the ordering of classes and characters in a given table, authorship and so on, the <code class="func">InfoText</code> (<a href="../../../doc/ref/chap12_mj.html#X871562FD7F982C12"><span class="RefLink">Reference: InfoText</span></a>) value of the table contains the information</p>
<dl>
<dt><strong class="Mark"><code class="code">origin: ATLAS of finite groups</code></strong></dt>
<dd><p>for <strong class="pkg">Atlas</strong> tables (see Section <a href="chap4_mj.html#X7F44BD4B79473085"><span class="RefLink">4.3</span></a>),</p>
</dd>
<dt><strong class="Mark"><code class="code">origin: Ostermann</code></strong></dt>
<dd><p>for tables contained in <a href="chapBib_mj.html#biBOst86">[Ost86]</a>,</p>
</dd>
<dt><strong class="Mark"><code class="code">origin: CAS library</code></strong></dt>
<dd><p>for any table of the <strong class="pkg">CAS</strong> table library that is contained neither in the <strong class="pkg">Atlas</strong> nor in <a href="chapBib_mj.html#biBOst86">[Ost86]</a>, and</p>
</dd>
<dt><strong class="Mark"><code class="code">origin: Hanrath library</code></strong></dt>
<dd><p>for tables contained in the microfiches in <a href="chapBib_mj.html#biBHP89">[HP89]</a>.</p>
</dd>
</dl>
<p>The <code class="func">InfoText</code> (<a href="../../../doc/ref/chap12_mj.html#X871562FD7F982C12"><span class="RefLink">Reference: InfoText</span></a>) value usually contains more detailed information, for example that the table in question is the character table of a maximal subgroup of an almost simple group. If the table was contained in the <strong class="pkg">CAS</strong> library then additional information may be available via the <code class="func">CASInfo</code> (<a href="chap4_mj.html#X786A80A279674E91"><span class="RefLink">4.4-1</span></a>) value.</p>
<p>If one is interested in the aspect of relations between the tables, i. e., the internal structure of the library of ordinary tables, the contents can be listed up the following way.</p>
<p>We have</p>
<ul>
<li><p>all <strong class="pkg">Atlas</strong> tables (see Section <a href="chap4_mj.html#X7F44BD4B79473085"><span class="RefLink">4.3</span></a>), i. e., the tables of the simple groups which are contained in the <strong class="pkg">Atlas</strong> of Finite Groups, and the tables of cyclic and bicyclic extensions of these groups,</p>
</li>
<li><p>most tables of maximal subgroups of sporadic simple groups (<em>not all</em> for the Monster group),</p>
</li>
<li><p>many tables of maximal subgroups of other <strong class="pkg">Atlas</strong> tables; the <code class="func">Maxes</code> (<a href="chap3_mj.html#X8150E63F7DBDF252"><span class="RefLink">3.7-1</span></a>) value for the table is set if all tables of maximal subgroups are available,</p>
</li>
<li><p>the tables of many Sylow <span class="SimpleMath">\(p\)</span>-normalizers of sporadic simple groups; this includes the tables printed in <a href="chapBib_mj.html#biBOst86">[Ost86]</a> except <span class="SimpleMath">\(J_4N2\)</span>, <span class="SimpleMath">\(Co_1N2\)</span>, <span class="SimpleMath">\(Fi_{22}N2\)</span>, but also other tables are available; more generally, several tables of normalizers of other radical <span class="SimpleMath">\(p\)</span>-subgroups are available, such as normalizers of defect groups of <span class="SimpleMath">\(p\)</span>-blocks,</p>
</li>
<li><p>some tables of element centralizers,</p>
</li>
<li><p>some tables of Sylow <span class="SimpleMath">\(p\)</span>-subgroups,</p>
</li>
<li><p>and a few other tables, e. g. <code class="code">W(F4)</code></p>
</li>
</ul>
<p><em>Note</em> that class fusions stored on library tables are not guaranteed to be compatible for any two subgroups of a group and their intersection, and they are not guaranteed to be consistent w. r. t. the composition of maps.</p>
<p><a id="X7AD048607A08C6FF" name="X7AD048607A08C6FF"></a></p>
<h5>4.1-2 <span class="Heading">Brauer Tables</span></h5>
<p>The library contains all tables of the <strong class="pkg">Atlas</strong> of Brauer Tables (<a href="chapBib_mj.html#biBJLPW95">[JLPW95]</a>), and many other Brauer tables of bicyclic extensions of simple groups which are known yet. The Brauer tables in the library contain the information</p>
<pre class="normal">
origin: modular ATLAS of finite groups
</pre>
<p>in their <code class="func">InfoText</code> (<a href="../../../doc/ref/chap12_mj.html#X871562FD7F982C12"><span class="RefLink">Reference: InfoText</span></a>) string.</p>
<p><a id="X81E3F9A384365282" name="X81E3F9A384365282"></a></p>
<h4>4.2 <span class="Heading">Generic Character Tables</span></h4>
<p>Generic character tables provide a means for writing down the character tables of all groups in a (usually infinite) series of similar groups, e. g., cyclic groups, or symmetric groups, or the general linear groups GL<span class="SimpleMath">\((2,q)\)</span> where <span class="SimpleMath">\(q\)</span> ranges over certain prime powers.</p>
<p>Let <span class="SimpleMath">\(\{ G_q | q \in I \}\)</span> be such a series, where <span class="SimpleMath">\(I\)</span> is an index set. The character table of one fixed member <span class="SimpleMath">\(G_q\)</span> could be computed using a function that takes <span class="SimpleMath">\(q\)</span> as only argument and constructs the table of <span class="SimpleMath">\(G_q\)</span>. It is, however, often desirable to compute not only the whole table but to access just one specific character, or to compute just one character value, without computing the whole character table.</p>
<p>For example, both the conjugacy classes and the irreducible characters of the symmetric group <span class="SimpleMath">\(S_n\)</span> are in bijection with the partitions of <span class="SimpleMath">\(n\)</span>. Thus for given <span class="SimpleMath">\(n\)</span> it makes sense to ask for the character corresponding to a particular partition, or just for its character value at another partition.</p>
<p>A generic character table in <strong class="pkg">GAP</strong> allows one such local evaluations. In this sense, <strong class="pkg">GAP</strong> can deal also with character tables that are too big to be computed and stored as a whole.</p>
<p>Currently the only operations for generic tables supported by <strong class="pkg">GAP</strong> are the specialisation of the parameter <span class="SimpleMath">\(q\)</span> in order to compute the whole character table of <span class="SimpleMath">\(G_q\)</span>, and local evaluation (see <code class="func">ClassParameters</code> (<a href="../../../doc/ref/chap71_mj.html#X8333E8038308947E"><span class="RefLink">Reference: ClassParameters</span></a>) for an example). <strong class="pkg">GAP</strong> does <em>not</em> support the computation of, e. g., generic scalar products.</p>
<p>While the numbers of conjugacy classes for the members of a series of groups are usually not bounded, there is always a fixed finite number of <em>types</em> (equivalence classes) of conjugacy classes; very often the equivalence relation is isomorphism of the centralizers of the representatives.</p>
<p>For each type <span class="SimpleMath">\(t\)</span> of classes and a fixed <span class="SimpleMath">\(q \in I\)</span>, a <em>parametrisation</em> of the classes in <span class="SimpleMath">\(t\)</span> is a function that assigns to each conjugacy class of <span class="SimpleMath">\(G_q\)</span> in <span class="SimpleMath">\(t\)</span> a <em>parameter</em> by which it is uniquely determined. Thus the classes are indexed by pairs <span class="SimpleMath">\([t,p_t]\)</span> consisting of a type <span class="SimpleMath">\(t\)</span> and a parameter <span class="SimpleMath">\(p_t\)</span> for that type.</p>
<p>For any generic table, there has to be a fixed number of types of irreducible characters of <span class="SimpleMath">\(G_q\)</span>, too. Like the classes, the characters of each type are parametrised.</p>
<p>In <strong class="pkg">GAP</strong>, the parametrisations of classes and characters for tables computed from generic tables is stored using the attributes <code class="func">ClassParameters</code> (<a href="../../../doc/ref/chap71_mj.html#X8333E8038308947E"><span class="RefLink">Reference: ClassParameters</span></a>) and <code class="func">CharacterParameters</code> (<a href="../../../doc/ref/chap71_mj.html#X8333E8038308947E"><span class="RefLink">Reference: CharacterParameters</span></a>).</p>
<p><a id="X78CD9A2D8680506B" name="X78CD9A2D8680506B"></a></p>
<h5>4.2-1 <span class="Heading">Available generic character tables</span></h5>
<p>Currently, generic tables of the following groups –in alphabetical order– are available in <strong class="pkg">GAP</strong>. (A list of the names of generic tables known to <strong class="pkg">GAP</strong> is <code class="code">LIBTABLE.GENERIC.firstnames</code>.) We list the function calls needed to get a specialized table, the generic table itself can be accessed by calling <code class="func">CharacterTable</code> (<a href="../../../doc/ref/chap71_mj.html#X7FCA7A7A822BDA33"><span class="RefLink">Reference: CharacterTable</span></a>) with the first argument only; for example, <code class="code">CharacterTable( "Cyclic" )</code> yields the generic table of cyclic groups.</p>
<dl>
<dt><strong class="Mark"><code class="code">CharacterTable( "Alternating", </code><span class="SimpleMath">\(n\)</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>alternating</em> group on <span class="SimpleMath">\(n\)</span> letters,</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "Cyclic", </code><span class="SimpleMath">\(n\)</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>cyclic</em> group of order <span class="SimpleMath">\(n\)</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "Dihedral", </code><span class="SimpleMath">\(2n\)</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>dihedral</em> group of order <span class="SimpleMath">\(2n\)</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "DoubleCoverAlternating", </code><span class="SimpleMath">\(n\)</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>Schur double cover of the alternating</em> group on <span class="SimpleMath">\(n\)</span> letters (see <a href="chapBib_mj.html#biBNoe02">[Noe02]</a>),</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "DoubleCoverSymmetric", </code><span class="SimpleMath">\(n\)</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>standard Schur double cover of the symmetric</em> group on <span class="SimpleMath">\(n\)</span> letters (see <a href="chapBib_mj.html#biBNoe02">[Noe02]</a>),</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "GL", 2, </code><span class="SimpleMath">\(q\)</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>general linear</em> group <code class="code">GL(2,</code><span class="SimpleMath">\(q\)</span><code class="code">)</code>, for a prime power <span class="SimpleMath">\(q\)</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "GU", 3, </code><span class="SimpleMath">\(q\)</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>general unitary</em> group <code class="code">GU(3,</code><span class="SimpleMath">\(q\)</span><code class="code">)</code>, for a prime power <span class="SimpleMath">\(q\)</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "P:Q", </code><span class="SimpleMath">\([ p, q ]\)</span><code class="code"> )</code> and
<code class="code">CharacterTable( "P:Q", </code><span class="SimpleMath">\([ p, q, k ]\)</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>Frobenius extension</em> of the nontrivial cyclic group of odd order <span class="SimpleMath">\(p\)</span> by the nontrivial cyclic group of order <span class="SimpleMath">\(q\)</span> where <span class="SimpleMath">\(q\)</span> divides <span class="SimpleMath">\(p_i-1\)</span> for all prime divisors <span class="SimpleMath">\(p_i\)</span> of <span class="SimpleMath">\(p\)</span>; if <span class="SimpleMath">\(p\)</span> is a prime power then <span class="SimpleMath">\(q\)</span> determines the group uniquely and thus the first version can be used, otherwise the action of the residue class of <span class="SimpleMath">\(k\)</span> modulo <span class="SimpleMath">\(p\)</span> is taken for forming orbits of length <span class="SimpleMath">\(q\)</span> each on the nonidentity elements of the group of order <span class="SimpleMath">\(p\)</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "PSL", 2, </code><span class="SimpleMath">\(q\)</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>projective special linear</em> group <code class="code">PSL(2,</code><span class="SimpleMath">\(q\)</span><code class="code">)</code>, for a prime power <span class="SimpleMath">\(q\)</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "SL", 2, </code><span class="SimpleMath">\(q\)</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>special linear</em> group <code class="code">SL(2,</code><span class="SimpleMath">\(q\)</span><code class="code">)</code>, for a prime power <span class="SimpleMath">\(q\)</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "SU", 3, </code><span class="SimpleMath">\(q\)</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>special unitary</em> group <code class="code">SU(3,</code><span class="SimpleMath">\(q\)</span><code class="code">)</code>, for a prime power <span class="SimpleMath">\(q\)</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "Suzuki", </code><span class="SimpleMath">\(q\)</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>Suzuki</em> group <code class="code">Sz(</code><span class="SimpleMath">\(q\)</span><code class="code">)</code> <span class="SimpleMath">\(= {}^2B_2(q)\)</span>, for <span class="SimpleMath">\(q\)</span> an odd power of <span class="SimpleMath">\(2\)</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "Symmetric", </code><span class="SimpleMath">\(n\)</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>symmetric</em> group on <span class="SimpleMath">\(n\)</span> letters,</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "WeylB", </code><span class="SimpleMath">\(n\)</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>Weyl</em> group of type <span class="SimpleMath">\(B_n\)</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "WeylD", </code><span class="SimpleMath">\(n\)</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>Weyl</em> group of type <span class="SimpleMath">\(D_n\)</span>.</p>
</dd>
</dl>
<p>In addition to the above calls that really use generic tables, the following calls to <code class="func">CharacterTable</code> (<a href="../../../doc/ref/chap71_mj.html#X7FCA7A7A822BDA33"><span class="RefLink">Reference: CharacterTable</span></a>) are to some extent <q>generic</q> constructions. But note that no local evaluation is possible in these cases, as no generic table object exists in <strong class="pkg">GAP</strong> that can be asked for local information.</p>
<dl>
<dt><strong class="Mark"><code class="code">CharacterTable( "Quaternionic", </code><span class="SimpleMath">\(4n\)</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>generalized quaternionic</em> group of order <span class="SimpleMath">\(4n\)</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTableWreathSymmetric( tbl, </code><span class="SimpleMath">\(n\)</span><code class="code"> )</code></strong></dt>
<dd><p>the character table of the wreath product of the group whose table is <code class="code">tbl</code> with the symmetric group on <span class="SimpleMath">\(n\)</span> letters, see <code class="func">CharacterTableWreathSymmetric</code> (<a href="../../../doc/ref/chap71_mj.html#X79B75C8582426BC5"><span class="RefLink">Reference: CharacterTableWreathSymmetric</span></a>).</p>
</dd>
</dl>
<p><a id="X78DA225F78F381C9" name="X78DA225F78F381C9"></a></p>
<h5>4.2-2 CharacterTableSpecialized</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CharacterTableSpecialized</code>( <var class="Arg">gentbl</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a record <var class="Arg">gentbl</var> representing a generic character table, and a parameter value <var class="Arg">q</var>, <code class="func">CharacterTableSpecialized</code> returns a character table object computed by evaluating <var class="Arg">gentbl</var> at <var class="Arg">q</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">c5:= CharacterTableSpecialized( CharacterTable( "Cyclic" ), 5 );</span>
CharacterTable( "C5" )
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( c5 );</span>
C5
5 1 1 1 1 1
1a 5a 5b 5c 5d
5P 1a 1a 1a 1a 1a
X.1 1 1 1 1 1
X.2 1 A B /B /A
X.3 1 B /A A /B
X.4 1 /B A /A B
X.5 1 /A /B B A
A = E(5)
B = E(5)^2
</pre></div>
<p>(Also <code class="code">CharacterTable( "Cyclic", 5 )</code> could have been used to construct the above table.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">HasClassParameters( c5 ); HasCharacterParameters( c5 );</span>
true
true
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassParameters( c5 ); CharacterParameters( c5 );</span>
[ [ 1, 0 ], [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 4 ] ]
[ [ 1, 0 ], [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 4 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassParameters( CharacterTable( "Symmetric", 3 ) );</span>
[ [ 1, [ 1, 1, 1 ] ], [ 1, [ 2, 1 ] ], [ 1, [ 3 ] ] ]
</pre></div>
<p>Here are examples for the <q>local evaluation</q> of generic character tables, first a character value of the cyclic group shown above, then a character value and a representative order of a symmetric group.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CharacterTable( "Cyclic" ).irreducibles[1][1]( 5, 2, 3 );</span>
E(5)
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "Symmetric" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl.irreducibles[1][1]( 5, [ 3, 2 ], [ 2, 2, 1 ] );</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl.orders[1]( 5, [ 2, 1, 1, 1 ] );</span>
2
</pre></div>
<p><a id="X7C2CB9E07990B63D" name="X7C2CB9E07990B63D"></a></p>
<h5>4.2-3 <span class="Heading">Components of generic character tables</span></h5>
<p>Any generic table in <strong class="pkg">GAP</strong> is represented by a record. The following components are supported for generic character table records.</p>
<dl>
<dt><strong class="Mark"><code class="code">centralizers</code></strong></dt>
<dd><p>list of functions, one for each class type <span class="SimpleMath">\(t\)</span>, with arguments <span class="SimpleMath">\(q\)</span> and <span class="SimpleMath">\(p_t\)</span>, returning the centralizer order of the class <span class="SimpleMath">\([t,p_t]\)</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">charparam</code></strong></dt>
<dd><p>list of functions, one for each character type <span class="SimpleMath">\(t\)</span>, with argument <span class="SimpleMath">\(q\)</span>, returning the list of character parameters of type <span class="SimpleMath">\(t\)</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">classparam</code></strong></dt>
<dd><p>list of functions, one for each class type <span class="SimpleMath">\(t\)</span>, with argument <span class="SimpleMath">\(q\)</span>, returning the list of class parameters of type <span class="SimpleMath">\(t\)</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">classtext</code></strong></dt>
<dd><p>list of functions, one for each class type <span class="SimpleMath">\(t\)</span>, with arguments <span class="SimpleMath">\(q\)</span> and <span class="SimpleMath">\(p_t\)</span>, returning a representative of the class with parameter <span class="SimpleMath">\([t,p_t]\)</span> (note that this element need <em>not</em> actually lie in the group in question, for example it may be a diagonal matrix but the characteristic polynomial in the group s irreducible),</p>
</dd>
<dt><strong class="Mark"><code class="code">domain</code></strong></dt>
<dd><p>function of <span class="SimpleMath">\(q\)</span> returning <code class="keyw">true</code> if <span class="SimpleMath">\(q\)</span> is a valid parameter, and <code class="keyw">false</code> otherwise,</p>
</dd>
<dt><strong class="Mark"><code class="code">identifier</code></strong></dt>
<dd><p>identifier string of the generic table,</p>
</dd>
<dt><strong class="Mark"><code class="code">irreducibles</code></strong></dt>
<dd><p>list of list of functions, in row <span class="SimpleMath">\(i\)</span> and column <span class="SimpleMath">\(j\)</span> the function of three arguments, namely <span class="SimpleMath">\(q\)</span> and the parameters <span class="SimpleMath">\(p_t\)</span> and <span class="SimpleMath">\(p_s\)</span> of the class type <span class="SimpleMath">\(t\)</span> and the character type <span class="SimpleMath">\(s\)</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">isGenericTable</code></strong></dt>
<dd><p>always <code class="keyw">true</code></p>
</dd>
<dt><strong class="Mark"><code class="code">libinfo</code></strong></dt>
<dd><p>record with components <code class="code">firstname</code> (<code class="func">Identifier</code> (<a href="../../../doc/ref/chap71_mj.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>) value of the table) and <code class="code">othernames</code> (list of other admissible names)</p>
</dd>
<dt><strong class="Mark"><code class="code">matrix</code></strong></dt>
<dd><p>function of <span class="SimpleMath">\(q\)</span> returning the matrix of irreducibles of <span class="SimpleMath">\(G_q\)</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">orders</code></strong></dt>
<dd><p>list of functions, one for each class type <span class="SimpleMath">\(t\)</span>, with arguments <span class="SimpleMath">\(q\)</span> and <span class="SimpleMath">\(p_t\)</span>, returning the representative order of elements of type <span class="SimpleMath">\(t\)</span> and parameter <span class="SimpleMath">\(p_t\)</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">powermap</code></strong></dt>
<dd><p>list of functions, one for each class type <span class="SimpleMath">\(t\)</span>, each with three arguments <span class="SimpleMath">\(q\)</span>, <span class="SimpleMath">\(p_t\)</span>, and <span class="SimpleMath">\(k\)</span>, returning the pair <span class="SimpleMath">\([s,p_s]\)</span> of type and parameter for the <span class="SimpleMath">\(k\)</span>-th power of the class with parameter <span class="SimpleMath">\([t,p_t]\)</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">size</code></strong></dt>
<dd><p>function of <span class="SimpleMath">\(q\)</span> returning the order of <span class="SimpleMath">\(G_q\)</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">specializedname</code></strong></dt>
<dd><p>function of <span class="SimpleMath">\(q\)</span> returning the <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71_mj.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>) value of the table of <span class="SimpleMath">\(G_q\)</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">text</code></strong></dt>
<dd><p>string informing about the generic table</p>
</dd>
</dl>
<p>In the specialized table, the <code class="func">ClassParameters</code> (<a href="../../../doc/ref/chap71_mj.html#X8333E8038308947E"><span class="RefLink">Reference: ClassParameters</span></a>) and <code class="func">CharacterParameters</code> (<a href="../../../doc/ref/chap71_mj.html#X8333E8038308947E"><span class="RefLink">Reference: CharacterParameters</span></a>) values are the lists of parameters <span class="SimpleMath">\([t,p_t]\)</span> of classes and characters, respectively.</p>
<p>If the <code class="code">matrix</code> component is present then its value implements a method to compute the complete table of small members <span class="SimpleMath">\(G_q\)</span> more efficiently than via local evaluation; this method will be called when the generic table is used to compute the whole character table for a given <span class="SimpleMath">\(q\)</span> (see <code class="func">CharacterTableSpecialized</code> (<a href="chap4_mj.html#X78DA225F78F381C9"><span class="RefLink">4.2-2</span></a>)).</p>
<p><a id="X7D693E9787073E30" name="X7D693E9787073E30"></a></p>
<h5>4.2-4 <span class="Heading">Example: The generic table of cyclic groups</span></h5>
<p>For the cyclic group <span class="SimpleMath">\(C_q = \langle x \rangle\)</span> of order <span class="SimpleMath">\(q\)</span>, there is one type of classes. The class parameters are integers <span class="SimpleMath">\(k \in \{ 0, \ldots, q-1 \}\)</span>, the class with parameter <span class="SimpleMath">\(k\)</span> consists of the group element <span class="SimpleMath">\(x^k\)</span>. Group order and centralizer orders are the identity function <span class="SimpleMath">\(q \mapsto q\)</span>, independent of the parameter <span class="SimpleMath">\(k\)</span>. The representative order function maps the parameter pair <span class="SimpleMath">\([q,k]\)</span> to <span class="SimpleMath">\(q / \gcd(q,k)\)</span>, which is the order of <span class="SimpleMath">\(x^k\)</span> in <span class="SimpleMath">\(C_q\)</span>; the <span class="SimpleMath">\(p\)</span>-th power map is the function mapping the triple <span class="SimpleMath">\((q,k,p)\)</span> to the parameter <span class="SimpleMath">\([1,(kp \bmod q)]\)</span>.</p>
<p>There is one type of characters, with parameters <span class="SimpleMath">\(l \in \{ 0, \ldots, q-1 \}\)</span>; for <span class="SimpleMath">\(e_q\)</span> a primitive complex <span class="SimpleMath">\(q\)</span>-th root of unity, the character values are <span class="SimpleMath">\(\chi_l(x^k) = e_q^{kl}\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Print( CharacterTable( "Cyclic" ), "\n" );</span>
rec(
centralizers := [ function ( n, k )
return n;
end ],
charparam := [ function ( n )
return [ 0 .. n - 1 ];
end ],
classparam := [ function ( n )
return [ 0 .. n - 1 ];
end ],
domain := <Category "(IsInt and IsPosRat)">,
identifier := "Cyclic",
irreducibles := [ [ function ( n, k, l )
return E( n ) ^ (k * l);
end ] ],
isGenericTable := true,
libinfo := rec(
firstname := "Cyclic",
othernames := [ ] ),
orders := [ function ( n, k )
return n / Gcd( n, k );
end ],
powermap := [ function ( n, k, pow )
return [ 1, k * pow mod n ];
end ],
size := function ( n )
return n;
end,
specializedname := function ( q )
return Concatenation( "C", String( q ) );
end,
text := "generic character table for cyclic groups" )
</pre></div>
<p><a id="X81231BCE79486FA3" name="X81231BCE79486FA3"></a></p>
<h5>4.2-5 <span class="Heading">Example: The generic table of the general linear group GL<span class="SimpleMath">\((2,q)\)</span>
</span></h5>
<p>We have four types <span class="SimpleMath">\(t_1, t_2, t_3, t_4\)</span> of classes, according to the rational canonical form of the elements. <span class="SimpleMath">\(t_1\)</span> describes scalar matrices, <span class="SimpleMath">\(t_2\)</span> nonscalar diagonal matrices, <span class="SimpleMath">\(t_3\)</span> companion matrices of <span class="SimpleMath">\((X - \rho)^2\)</span> for nonzero elements <span class="SimpleMath">\(\rho \in F_q\)</span>, and <span class="SimpleMath">\(t_4\)</span> companion matrices of irreducible polynomials of degree <span class="SimpleMath">\(2\)</span> over <span class="SimpleMath">\(F_q\)</span>.</p>
<p>The sets of class parameters of the types are in bijection with nonzero elements in <span class="SimpleMath">\(F_q\)</span> for <span class="SimpleMath">\(t_1\)</span> and <span class="SimpleMath">\(t_3\)</span>, with the set</p>
<p class="center">\[
\{ \{ \rho, \tau \};
\rho, \tau \in F_q, \rho ≠ 0, \tau ≠ 0, \rho ≠ \tau \}
\]</p>
<p>for <span class="SimpleMath">\(t_2\)</span>, and with the set <span class="SimpleMath">\(\{ \{ \epsilon, \epsilon^q \}; \epsilon \in F_{{q^2}} \setminus F_q \}\)</span> for <span class="SimpleMath">\(t_4\)</span>.</p>
<p>The centralizer order functions are <span class="SimpleMath">\(q \mapsto (q^2-1)(q^2-q)\)</span> for type <span class="SimpleMath">\(t_1\)</span>, <span class="SimpleMath">\(q \mapsto (q-1)^2\)</span> for type <span class="SimpleMath">\(t_2\)</span>, <span class="SimpleMath">\(q \mapsto q(q-1)\)</span> for type <span class="SimpleMath">\(t_3\)</span>, and <span class="SimpleMath">\(q \mapsto q^2-1\)</span> for type <span class="SimpleMath">\(t_4\)</span>.</p>
<p>The representative order function of <span class="SimpleMath">\(t_1\)</span> maps <span class="SimpleMath">\((q, \rho)\)</span> to the order of <span class="SimpleMath">\(\rho\)</span> in <span class="SimpleMath">\(F_q\)</span>, that of <span class="SimpleMath">\(t_2\)</span> maps <span class="SimpleMath">\((q, \{ \rho, \tau \})\)</span> to the least common multiple of the orders of <span class="SimpleMath">\(\rho\)</span> and <span class="SimpleMath">\(\tau\)</span>.</p>
<p>The file contains something similar to the following table.</p>
<div class="example"><pre>
rec(
identifier := "GL2",
specializedname := ( q -> Concatenation( "GL(2,", String(q), ")" ) ),
size := ( q -> (q^2-1)*(q^2-q) ),
text := "generic character table of GL(2,q), see Robert Steinberg: ...",
centralizers := [ function( q, k ) return (q^2-1) * (q^2-q); end,
..., ..., ... ],
classparam := [ ( q -> [ 0 .. q-2 ] ), ..., ..., ... ],
charparam := [ ( q -> [ 0 .. q-2 ] ), ..., ..., ... ],
powermap := [ function( q, k, pow ) return [ 1, (k*pow) mod (q-1) ]; end,
..., ..., ... ],
orders:= [ function( q, k ) return (q-1)/Gcd( q-1, k ); end,
..., ..., ... ],
irreducibles := [ [ function( q, k, l ) return E(q-1)^(2*k*l); end,
..., ..., ... ],
[ ..., ..., ..., ... ],
[ ..., ..., ..., ... ],
[ ..., ..., ..., ... ] ],
classtext := [ ..., ..., ..., ... ],
domain := IsPrimePowerInt,
isGenericTable := true )
</pre></div>
<p><a id="X7F44BD4B79473085" name="X7F44BD4B79473085"></a></p>
<h4>4.3 <span class="Heading"><strong class="pkg">Atlas</strong> Tables</span></h4>
<p>The <strong class="pkg">GAP</strong> character table library contains all character tables of bicyclic extensions of simple groups that are included in the <strong class="pkg">Atlas</strong> of Finite Groups (<a href="chapBib_mj.html#biBCCN85">[CCN+85]</a>, from now on called <strong class="pkg">Atlas</strong>), and the Brauer tables contained in the <strong class="pkg">Atlas</strong> of Brauer Characters (<a href="chapBib_mj.html#biBJLPW95">[JLPW95]</a>).</p>
<p>These tables have the information</p>
<pre class="normal">
origin: ATLAS of finite groups
</pre>
<p>or</p>
<pre class="normal">
origin: modular ATLAS of finite groups
</pre>
<p>in their <code class="func">InfoText</code> (<a href="../../../doc/ref/chap12_mj.html#X871562FD7F982C12"><span class="RefLink">Reference: InfoText</span></a>) value, they are simply called <strong class="pkg">Atlas</strong> tables further on.</p>
<p>The property <code class="func">IsAtlasCharacterTable</code> (<a href="chap3_mj.html#X867813EB79DC6953"><span class="RefLink">3.7-6</span></a>) describes which character tables are <strong class="pkg">Atlas</strong> tables.</p>
<p>For displaying <strong class="pkg">Atlas</strong> tables with the row labels used in the <strong class="pkg">Atlas</strong>, or for displaying decomposition matrices, see <code class="func">LaTeXStringDecompositionMatrix</code> (<a href="../../../doc/ref/chap71_mj.html#X83EC921380AF9B3B"><span class="RefLink">Reference: LaTeXStringDecompositionMatrix</span></a>) and <code class="func">AtlasLabelsOfIrreducibles</code> (<a href="chap4_mj.html#X7ADC9DC980CF0685"><span class="RefLink">4.3-6</span></a>).</p>
<p>In addition to the information given in Chapters 6 to 8 of the <strong class="pkg">Atlas</strong> which tell you how to read the printed tables, there are some rules relating these to the corresponding <strong class="pkg">GAP</strong> tables.</p>
<p><a id="X7CC608CD8690F9B1" name="X7CC608CD8690F9B1"></a></p>
<h5>4.3-1 <span class="Heading">Improvements to the <strong class="pkg">Atlas</strong></span></h5>
<p>For the <strong class="pkg">GAP</strong> Character Table Library not the printed versions of the <strong class="pkg">Atlas</strong> of Finite Groups and the <strong class="pkg">Atlas</strong> of Brauer Characters are relevant but the revised versions given by the currently three lists of improvements that are maintained by Simon Norton. The first such list is contained in <a href="chapBib_mj.html#biBBN95">[BN95]</a>, and is printed in the Appendix of <a href="chapBib_mj.html#biBJLPW95">[JLPW95]</a>; it contains the improvements that had been known until the <q><strong class="pkg">Atlas</strong> of Brauer Characters</q> was published. The second list contains the improvements to the <strong class="pkg">Atlas</strong> of Finite Groups that were found since the publication of <a href="chapBib_mj.html#biBJLPW95">[JLPW95]</a>. It can be found in the internet, an HTML version at</p>
<p><span class="URL"><a href="http://web.mat.bham.ac.uk/atlas/html/atlasmods.html">http://web.mat.bham.ac.uk/atlas/html/atlasmods.html</a></span></p>
<p>and a DVI version at</p>
<p><span class="URL"><a href="http://web.mat.bham.ac.uk/atlas/html/atlasmods.dvi">http://web.mat.bham.ac.uk/atlas/html/atlasmods.dvi</a></span>.</p>
<p>The third list contains the improvements to the <strong class="pkg">Atlas</strong> of Brauer Characters, HTML and PDF versions can be found in the internet at</p>
<p><span class="URL"><a href="https://www.math.rwth-aachen.de/~MOC/ABCerr.html">https://www.math.rwth-aachen.de/~MOC/ABCerr.html</a></span></p>
<p>and</p>
<p><span class="URL"><a href="https://www.math.rwth-aachen.de/~MOC/ABCerr.pdf">https://www.math.rwth-aachen.de/~MOC/ABCerr.pdf</a></span>,</p>
<p>respectively.</p>
<p>Also some tables are regarded as <strong class="pkg">Atlas</strong> tables that are not printed in the <strong class="pkg">Atlas</strong> but available in <strong class="pkg">Atlas</strong> format, according to the lists of improvements mentioned above. Currently these are the tables related to <span class="SimpleMath">\(L_2(49)\)</span>, <span class="SimpleMath">\(L_2(81)\)</span>, <span class="SimpleMath">\(L_6(2)\)</span>, <span class="SimpleMath">\(O_8^-(3)\)</span>, <span class="SimpleMath">\(O_8^+(3)\)</span>, <span class="SimpleMath">\(S_{10}(2)\)</span>, and <span class="SimpleMath">\({}^2E_6(2).3\)</span>.</p>
<p><a id="X7FED949A86575949" name="X7FED949A86575949"></a></p>
<h5>4.3-2 <span class="Heading">Power Maps</span></h5>
<p>For the tables of <span class="SimpleMath">\(3.McL\)</span>, <span class="SimpleMath">\(3_2.U_4(3)\)</span> and its covers, and <span class="SimpleMath">\(3_2.U_4(3).2_3\)</span> and its covers, the power maps are not uniquely determined by the information from the <strong class="pkg">Atlas</strong> but determined only up to matrix automorphisms (see <code class="func">MatrixAutomorphisms</code> (<a href="../../../doc/ref/chap71_mj.html#X84353BB884AF0365"><span class="RefLink">Reference: MatrixAutomorphisms</span></a>)) of the irreducible characters. In these cases, the first possible map according to lexicographical ordering was chosen, and the automorphisms are listed in the <code class="func">InfoText</code> (<a href="../../../doc/ref/chap12_mj.html#X871562FD7F982C12"><span class="RefLink">Reference: InfoText</span></a>) strings of the tables.</p>
<p><a id="X824823F47BB6AD6C" name="X824823F47BB6AD6C"></a></p>
<h5>4.3-3 <span class="Heading">Projective Characters and Projections</span></h5>
<p>If <span class="SimpleMath">\(G\)</span> (or <span class="SimpleMath">\(G.a\)</span>) has a nontrivial Schur multiplier then the attribute <code class="func">ProjectivesInfo</code> (<a href="chap3_mj.html#X82DC2E7779322DA8"><span class="RefLink">3.7-2</span></a>) of the <strong class="pkg">GAP</strong> table object of <span class="SimpleMath">\(G\)</span> (or <span class="SimpleMath">\(G.a\)</span>) is set; the <code class="code">chars</code> component of the record in question is the list of values lists of those faithful projective irreducibles that are printed in the <strong class="pkg">Atlas</strong> (so-called <em>proxy character</em>), and the <code class="code">map</code> component lists the positions of columns in the covering for which the column is printed in the <strong class="pkg">Atlas</strong> (a so-called <em>proxy class</em>, this preimage is denoted by <span class="SimpleMath">\(g_0\)</span> in Chapter 7, Section 14 of the <strong class="pkg">Atlas</strong>).</p>
<p><a id="X78732CDF85FB6774" name="X78732CDF85FB6774"></a></p>
<h5>4.3-4 <span class="Heading">Tables of Isoclinic Groups</span></h5>
<p>As described in Chapter 6, Section 7 and in Chapter 7, Section 18 of the <strong class="pkg">Atlas</strong>, there exist two (often nonisomorphic) groups of structure <span class="SimpleMath">\(2.G.2\)</span> for a simple group <span class="SimpleMath">\(G\)</span>, which are isoclinic. The table in the <strong class="pkg">GAP</strong> Character Table Library is the one printed in the <strong class="pkg">Atlas</strong>, the table of the isoclinic variant can be constructed using <code class="func">CharacterTableIsoclinic</code> (<a href="../../../doc/ref/chap71_mj.html#X85BE46F784B83938"><span class="RefLink">Reference: CharacterTableIsoclinic</span></a>).</p>
<p><a id="X7C4B91CD84D5CDCC" name="X7C4B91CD84D5CDCC"></a></p>
<h5>4.3-5 <span class="Heading">Ordering of Characters and Classes</span></h5>
<p>(Throughout this section, <span class="SimpleMath">\(G\)</span> always means the simple group involved.)</p>
<ol>
<li><p>For <span class="SimpleMath">\(G\)</span> itself, the ordering of classes and characters in the <strong class="pkg">GAP</strong> table coincides with the one in the <strong class="pkg">Atlas</strong>.</p>
</li>
<li><p>For an automorphic extension <span class="SimpleMath">\(G.a\)</span>, there are three types of characters.</p>
<ul>
<li><p>If a character <span class="SimpleMath">\(\chi\)</span> of <span class="SimpleMath">\(G\)</span> extends to <span class="SimpleMath">\(G.a\)</span> then the different extensions <span class="SimpleMath">\(\chi^0, \chi^1, \ldots, \chi^{{a-1}}\)</span> are consecutive in the table of <span class="SimpleMath">\(G.a\)</span> (see <a href="chapBib_mj.html#biBCCN85">[CCN+85, Chapter 7, Section 16]</a>).</p>
</li>
<li><p>If some characters of <span class="SimpleMath">\(G\)</span> fuse to give a single character of <span class="SimpleMath">\(G.a\)</span> then the position of that character in the table of <span class="SimpleMath">\(G.a\)</span> is given by the position of the first involved character of <span class="SimpleMath">\(G\)</span>.</p>
</li>
<li><p>If both extension and fusion occur for a character then the resulting characters are consecutive in the table of <span class="SimpleMath">\(G.a\)</span>, and each replaces the first involved character of <span class="SimpleMath">\(G\)</span>.</p>
</li>
</ul>
</li>
<li><p>Similarly, there are different types of classes for an automorphic extension <span class="SimpleMath">\(G.a\)</span>, as follows.</p>
<ul>
<li><p>If some classes collapse then the resulting class replaces the first involved class of <span class="SimpleMath">\(G\)</span>.</p>
</li>
<li><p>For <span class="SimpleMath">\(a > 2\)</span>, any proxy class and its algebraic conjugates that are not printed in the <strong class="pkg">Atlas</strong> are consecutive in the table of <span class="SimpleMath">\(G.a\)</span>; if more than two classes of <span class="SimpleMath">\(G.a\)</span> have the same proxy class (the only case that actually occurs is for <span class="SimpleMath">\(a = 5\)</span>) then the ordering of non-printed classes is the natural one of corresponding Galois conjugacy operators <span class="SimpleMath">\(*k\)</span> (see <a href="chapBib_mj.html#biBCCN85">[CCN+85, Chapter 7, Section 19]</a>).</p>
</li>
<li><p>For <span class="SimpleMath">\(a_1\)</span>, <span class="SimpleMath">\(a_2\)</span> dividing <span class="SimpleMath">\(a\)</span> such that <span class="SimpleMath">\(a_1 \leq a_2\)</span>, the classes of <span class="SimpleMath">\(G.a_1\)</span> in <span class="SimpleMath">\(G.a\)</span> precede the classes of <span class="SimpleMath">\(G.a_2\)</span> not contained in <span class="SimpleMath">\(G.a_1\)</span>. This ordering is the same as in the <strong class="pkg">Atlas</strong>, with the only exception <span class="SimpleMath">\(U_3(8).6\)</span>.</p>
</li>
</ul>
</li>
<li><p>For a central extension <span class="SimpleMath">\(M.G\)</span>, there are two different types of characters, as follows.</p>
<ul>
<li><p>Each character can be regarded as a faithful character of a factor group <span class="SimpleMath">\(m.G\)</span>, where <span class="SimpleMath">\(m\)</span> divides <span class="SimpleMath">\(M\)</span>. Characters with the same kernel are consecutive as in the <strong class="pkg">Atlas</strong>, the ordering of characters with different kernels is given by the order of precedence <span class="SimpleMath">\(1, 2, 4, 3, 6, 12\)</span> for the different values of <span class="SimpleMath">\(m\)</span>.</p>
</li>
<li><p>If <span class="SimpleMath">\(m > 2\)</span>, a faithful character of <span class="SimpleMath">\(m.G\)</span> that is printed in the <strong class="pkg">Atlas</strong> (a so-called <em>proxy character</em>) represents two or more Galois conjugates. In each <strong class="pkg">Atlas</strong> table in <strong class="pkg">GAP</strong>, a proxy character always precedes the non-printed characters with this proxy. The case <span class="SimpleMath">\(m = 12\)</span> is the only one that actually occurs where more than one character for a proxy is not printed. In this case, the non-printed characters are ordered according to the corresponding Galois conjugacy operators <span class="SimpleMath">\(*5\)</span>, <span class="SimpleMath">\(*7\)</span>, <span class="SimpleMath">\(*11\)</span> (in this order).</p>
</li>
</ul>
</li>
<li><p>For the classes of a central extension we have the following.</p>
<ul>
<li><p>The preimages of a <span class="SimpleMath">\(G\)</span>-class in <span class="SimpleMath">\(M.G\)</span> are subsequent, the ordering is the same as that of the lifting order rows in <a href="chapBib_mj.html#biBCCN85">[CCN+85, Chapter 7, Section 7]</a>.</p>
</li>
<li><p>The primitive roots of unity chosen to represent the generating central element (i. e., the element in the second class of the <strong class="pkg">GAP</strong> table) are <code class="code">E(3)</code>, <code class="code">E(4)</code>, <code class="code">E(6)^5</code> (<code class="code">= E(2)*E(3)</code>), and <code class="code">E(12)^7</code> (<code class="code">= E(3)*E(4)</code>), for <span class="SimpleMath">\(m = 3\)</span>, <span class="SimpleMath">\(4\)</span>, <span class=" | | |
