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<p><a id="X81C250407B91443F" name="X81C250407B91443F"></a></p>
<div class="ChapSects"><a href="chap6.html#X81C250407B91443F">6 <span class="Heading">Interfaces to Other Data Formats for Character Tables</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7AFAF3527D86A210">6.1 <span class="Heading">Interface to the <strong class="pkg">CAS</strong> System</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X841537DA7E495B73">6.1-1 CASString</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X833464CD7B5FC8E3">6.2 <span class="Heading">Interface to the <strong class="pkg">MOC</strong> System</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X842957217E3697A1">6.2-1 MAKElb11</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X832005B37F0A2872">6.2-2 MOCTable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7FBC528C7818814C">6.2-3 MOCString</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7F17B82879C8F953">6.2-4 ScanMOC</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7FBA67C585BEBFE5">6.2-5 GAPChars</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7BF1B6BF83BDD27E">6.2-6 MOCChars</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X85BBAB537B4B72F5">6.3 <span class="Heading">Interface to <strong class="pkg">GAP</strong> 3</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7DB321DE80046931">6.3-1 GAP3CharacterTableScan</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7B2C53137D1227FE">6.3-2 GAP3CharacterTableString</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7B2725D2789A9D85">6.3-3 GAP3CharacterTableData</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7806073380076800">6.4 <span class="Heading">Interface to the Cambridge Format</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X78A84889878D98BB">6.4-1 CambridgeMaps</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X8209063C813240E5">6.4-2 StringOfCambridgeFormat</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X863CB6737FF9B740">6.5 <span class="Heading">Interface to the <strong class="pkg">MAGMA</strong> System</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X79D160BD7ECA6D2F">6.5-1 BosmaBase</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X8690C41E7F8EEDF9">6.5-2 GAPTableOfMagmaFile</a></span>
</div></div>
</div>
<h3>6 <span class="Heading">Interfaces to Other Data Formats for Character Tables</span></h3>
<p>This chapter describes data formats for character tables that can be read or created by <strong class="pkg">GAP</strong>. Currently these are the formats used by</p>
<ul>
<li><p>the <strong class="pkg">CAS</strong> system (see <a href="chap6.html#X7AFAF3527D86A210"><span class="RefLink">6.1</span></a>),</p>
</li>
<li><p>the <strong class="pkg">MOC</strong> system (see <a href="chap6.html#X833464CD7B5FC8E3"><span class="RefLink">6.2</span></a>),</p>
</li>
<li><p><strong class="pkg">GAP</strong> 3 (see <a href="chap6.html#X85BBAB537B4B72F5"><span class="RefLink">6.3</span></a>),</p>
</li>
<li><p>the so-called <em>Cambridge format</em> (see <a href="chap6.html#X7806073380076800"><span class="RefLink">6.4</span></a>), and</p>
</li>
<li><p>the <strong class="pkg">MAGMA</strong> system (see <a href="chap6.html#X863CB6737FF9B740"><span class="RefLink">6.5</span></a>).</p>
</li>
</ul>
<p><a id="X7AFAF3527D86A210" name="X7AFAF3527D86A210"></a></p>
<h4>6.1 <span class="Heading">Interface to the <strong class="pkg">CAS</strong> System</span></h4>
<p>The interface to <strong class="pkg">CAS</strong> (see <a href="chapBib.html#biBNPP84">[NPP84]</a>) is thought just for printing the <strong class="pkg">CAS</strong> data to a file. The function <code class="func">CASString</code> (<a href="chap6.html#X841537DA7E495B73"><span class="RefLink">6.1-1</span></a>) is available mainly in order to document the data format. <em>Reading</em> <strong class="pkg">CAS</strong> tables is not supported; note that the tables contained in the <strong class="pkg">CAS</strong> Character Table Library have been migrated to <strong class="pkg">GAP</strong> using a few <code class="code">sed</code> scripts and <code class="code">C</code> programs.</p>
<p><a id="X841537DA7E495B73" name="X841537DA7E495B73"></a></p>
<h5>6.1-1 CASString</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CASString</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>is a string that encodes the <strong class="pkg">CAS</strong> library format of the character table <var class="Arg">tbl</var>. This string can be printed to a file which then can be read into the <strong class="pkg">CAS</strong> system using its <code class="code">get</code> command (see <a href="chapBib.html#biBNPP84">[NPP84]</a>).</p>
<p>The used line length is the first entry in the list returned by <code class="func">SizeScreen</code> (<span class="RefLink">Reference: SizeScreen</span>).</p>
<p>Only the known values of the following attributes are used. <code class="func">ClassParameters</code> (<span class="RefLink">Reference: ClassParameters</span>) (for partitions only), <code class="func">ComputedClassFusions</code> (<span class="RefLink">Reference: ComputedClassFusions</span>), <code class="func">ComputedIndicators</code> (<span class="RefLink">Reference: ComputedIndicators</span>), <code class="func">ComputedPowerMaps</code> (<span class="RefLink">Reference: ComputedPowerMaps</span>), <code class="func">ComputedPrimeBlocks</code> (<span class="RefLink">Reference: ComputedPrimeBlockss</span>), <code class="func">Identifier</code> (<span class="RefLink">Reference: Identifier (for character tables)</span>), <code class="func">InfoText</code> (<span class="RefLink">Reference: InfoText</span>), <code class="func">Irr</code> (<span class="RefLink">Reference: Irr</span>), <code class="func">OrdersClassRepresentatives</code> (<span class="RefLink">Reference: OrdersClassRepresentatives</span>), <code class="func">Size</code> (<span class="RefLink">Reference: Size</span>), <code class="func">SizesCentralizers</code> (<span class="RefLink">Reference: SizesCentralisers</span>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Print( CASString( CharacterTable( "Cyclic", 2 ) ), "\n" );</span>
'C2'
00/00/00. 00.00.00.
(2,2,0,2,-1,0)
text:
(#computed using generic character table for cyclic groups#),
order=2,
centralizers:(
2,2
),
reps:(
1,2
),
powermap:2(
1,1
),
characters:
(1,1
,0:0)
(1,-1
,0:0);
/// converted from GAP
</pre></div>
<p><a id="X833464CD7B5FC8E3" name="X833464CD7B5FC8E3"></a></p>
<h4>6.2 <span class="Heading">Interface to the <strong class="pkg">MOC</strong> System</span></h4>
<p>The interface to <strong class="pkg">MOC</strong> (see <a href="chapBib.html#biBHJLP92">[HJLP]</a>) can be used to print <strong class="pkg">MOC</strong> input. Additionally it provides an alternative representation of (virtual) characters.</p>
<p>The <strong class="pkg">MOC</strong> 3 code of a <span class="SimpleMath">5</span> digit number in <strong class="pkg">MOC</strong> 2 code is given by the following list. (Note that the code must contain only lower case letters.)</p>
<pre class="normal">
ABCD for 0ABCD
a for 10000
b for 10001 k for 20001
c for 10002 l for 20002
d for 10003 m for 20003
e for 10004 n for 20004
f for 10005 o for 20005
g for 10006 p for 20006
h for 10007 q for 20007
i for 10008 r for 20008
j for 10009 s for 20009
tAB for 100AB
uAB for 200AB
vABCD for 1ABCD
wABCD for 2ABCD
yABC for 30ABC
z for 31000
</pre>
<p><em>Note</em> that any long number in <strong class="pkg">MOC</strong> 2 format is divided into packages of length <span class="SimpleMath">4</span>, the first (!) one filled with leading zeros if necessary. Such a number with decimals <span class="SimpleMath">d_1, d_2, ..., d_{4n+k}</span> is the sequence <span class="SimpleMath">0 d_1 d_2 d_3 d_4 ... 0 d_{4n-3} d_{4n-2} d_{4n-1} d_4n d_{4n+1} ... d_{4n+k}</span> where <span class="SimpleMath">0 ≤ k ≤ 3</span>, the first digit of <span class="SimpleMath">x</span> is <span class="SimpleMath">1</span> if the number is positive and <span class="SimpleMath">2</span> if the number is negative, and then follow <span class="SimpleMath">(4-k)</span> zeros.</p>
<p>Details about the <strong class="pkg">MOC</strong> system are explained in <a href="chapBib.html#biBHJLP92">[HJLP]</a>, a brief description can be found in <a href="chapBib.html#biBLP91">[LP91]</a>.</p>
<p><a id="X842957217E3697A1" name="X842957217E3697A1"></a></p>
<h5>6.2-1 MAKElb11</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MAKElb11</code>( <var class="Arg">listofns</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a list <var class="Arg">listofns</var> of positive integers, <code class="func">MAKElb11</code> prints field information for all number fields with conductor in this list.</p>
<p>The output of <code class="func">MAKElb11</code> is used by the <strong class="pkg">MOC</strong> system; Calling <code class="code">MAKElb11( [ 3 .. 189 ] )</code> will print something very similar to Richard Parker's file <code class="file">lb11</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">MAKElb11( [ 3, 4 ] );</span>
3 2 0 1 0
4 2 0 1 0
</pre></div>
<p><a id="X832005B37F0A2872" name="X832005B37F0A2872"></a></p>
<h5>6.2-2 MOCTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MOCTable</code>( <var class="Arg">gaptbl</var>[, <var class="Arg">basicset</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">MOCTable</code> returns the <strong class="pkg">MOC</strong> table record of the <strong class="pkg">GAP</strong> character table <var class="Arg">gaptbl</var>.</p>
<p>The one argument version can be used only if <var class="Arg">gaptbl</var> is an ordinary (<span class="SimpleMath">G.0</span>) table. For Brauer (<span class="SimpleMath">G.p</span>) tables, one has to specify a basic set <var class="Arg">basicset</var> of ordinary irreducibles. <var class="Arg">basicset</var> must then be a list of positions of the basic set characters in the <code class="func">Irr</code> (<span class="RefLink">Reference: Irr</span>) list of the ordinary table of <var class="Arg">gaptbl</var>.</p>
<p>The result is a record that contains the information of <var class="Arg">gaptbl</var> in a format similar to the <strong class="pkg">MOC</strong> 3 format. This record can, e. g., easily be printed out or be used to print out characters using <code class="func">MOCString</code> (<a href="chap6.html#X7FBC528C7818814C"><span class="RefLink">6.2-3</span></a>).</p>
<p>The components of the result are</p>
<dl>
<dt><strong class="Mark"><code class="code">identifier</code></strong></dt>
<dd><p>the string <code class="code">MOCTable( </code><span class="SimpleMath">name</span><code class="code"> )</code> where <span class="SimpleMath">name</span> is the <code class="func">Identifier</code> (<span class="RefLink">Reference: Identifier (for character tables)</span>) value of <var class="Arg">gaptbl</var>,</p>
</dd>
<dt><strong class="Mark"><code class="code">GAPtbl</code></strong></dt>
<dd><p><var class="Arg">gaptbl</var>,</p>
</dd>
<dt><strong class="Mark"><code class="code">prime</code></strong></dt>
<dd><p>the characteristic of the field (label <code class="code">30105</code> in <strong class="pkg">MOC</strong>),</p>
</dd>
<dt><strong class="Mark"><code class="code">centralizers</code></strong></dt>
<dd><p>centralizer orders for cyclic subgroups (label <code class="code">30130</code>)</p>
</dd>
<dt><strong class="Mark"><code class="code">orders</code></strong></dt>
<dd><p>element orders for cyclic subgroups (label <code class="code">30140</code>)</p>
</dd>
<dt><strong class="Mark"><code class="code">fieldbases</code></strong></dt>
<dd><p>at position <span class="SimpleMath">i</span> the Parker basis of the number field generated by the character values of the <span class="SimpleMath">i</span>-th cyclic subgroup. The length of <code class="code">fieldbases</code> is equal to the value of label <code class="code">30110</code> in <strong class="pkg">MOC</strong>.</p>
</dd>
<dt><strong class="Mark"><code class="code">cycsubgps</code></strong></dt>
<dd><p><code class="code">cycsubgps[i] = j</code> means that class <code class="code">i</code> of the <strong class="pkg">GAP</strong> table belongs to the <code class="code">j</code>-th cyclic subgroup of the <strong class="pkg">GAP</strong> table,</p>
</dd>
<dt><strong class="Mark"><code class="code">repcycsub</code></strong></dt>
<dd><p><code class="code">repcycsub[j] = i</code> means that class <code class="code">i</code> of the <strong class="pkg">GAP</strong> table is the representative of the <code class="code">j</code>-th cyclic subgroup of the <strong class="pkg">GAP</strong> table. <em>Note</em> that the representatives of <strong class="pkg">GAP</strong> table and <strong class="pkg">MOC</strong> table need not agree!</p>
</dd>
<dt><strong class="Mark"><code class="code">galconjinfo</code></strong></dt>
<dd><p>a list <span class="SimpleMath">[ r_1, c_1, r_2, c_2, ..., r_n, c_n ]</span> which means that the <span class="SimpleMath">i</span>-th class of the <strong class="pkg">GAP</strong> table is the <span class="SimpleMath">c_i</span>-th conjugate of the representative of the <span class="SimpleMath">r_i</span>-th cyclic subgroup on the <strong class="pkg">MOC</strong> table. (This is used to translate back to <strong class="pkg">GAP</strong> format, stored under label <code class="code">30160</code>)</p>
</dd>
<dt><strong class="Mark"><code class="code">30170</code></strong></dt>
<dd><p>(power maps) for each cyclic subgroup (except the trivial one) and each prime divisor of the representative order store four values, namely the number of the subgroup, the power, the number of the cyclic subgroup containing the image, and the power to which the representative must be raised to yield the image class. (This is used only to construct the <code class="code">30230</code> power map/embedding information.) In <code class="code">30170</code> only a list of lists (one for each cyclic subgroup) of all these values is stored, it will not be used by <strong class="pkg">GAP</strong>.</p>
</dd>
<dt><strong class="Mark"><code class="code">tensinfo</code></strong></dt>
<dd><p>tensor product information, used to compute the coefficients of the Parker base for tensor products of characters (label <code class="code">30210</code> in <strong class="pkg">MOC</strong>). For a field with vector space basis <span class="SimpleMath">(v_1, v_2, ..., v_n)</span>, the tensor product information of a cyclic subgroup in <strong class="pkg">MOC</strong> (as computed by <code class="code">fct</code>) is either <span class="SimpleMath">1</span> (for rational classes) or a sequence</p>
<p class="pcenter">n x_1,1 y_1,1 z_1,1 x_1,2 y_1,2 z_1,2 ... x_1,m_1 y_1,m_1 z_1,m_1 0 x_2,1 y_2,1 z_2,1 x_2,2 y_2,2 z_2,2 ... x_2,m_2 y_2,m_2 z_2,m_2 0 ... z_n,m_n 0</p>
<p>which means that the coefficient of <span class="SimpleMath">v_k</span> in the product</p>
<p class="pcenter">( ∑_i=1^n a_i v_i ) ( ∑_j=1^n b_j v_j )</p>
<p>is equal to</p>
<p class="pcenter">∑_i=1^m_k x_k,i a_y_k,i} b_z_k,i} .</p>
<p>On a <strong class="pkg">MOC</strong> table in <strong class="pkg">GAP</strong>, the <code class="code">tensinfo</code> component is a list of lists, each containing exactly the sequence mentioned above.</p>
</dd>
<dt><strong class="Mark"><code class="code">invmap</code></strong></dt>
<dd><p>inverse map to compute complex conjugate characters, label <code class="code">30220</code> in <strong class="pkg">MOC</strong>.</p>
</dd>
<dt><strong class="Mark"><code class="code">powerinfo</code></strong></dt>
<dd><p>field embeddings for <span class="SimpleMath">p</span>-th symmetrizations, <span class="SimpleMath">p</span> a prime integer not larger than the largest element order, label <code class="code">30230</code> in <strong class="pkg">MOC</strong>.</p>
</dd>
<dt><strong class="Mark"><code class="code">30900</code></strong></dt>
<dd><p>basic set of restricted ordinary irreducibles in the case of nonzero characteristic, all ordinary irreducibles otherwise.</p>
</dd>
</dl>
<p><a id="X7FBC528C7818814C" name="X7FBC528C7818814C"></a></p>
<h5>6.2-3 MOCString</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MOCString</code>( <var class="Arg">moctbl</var>[, <var class="Arg">chars</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">moctbl</var> be a <strong class="pkg">MOC</strong> table record, as returned by <code class="func">MOCTable</code> (<a href="chap6.html#X832005B37F0A2872"><span class="RefLink">6.2-2</span></a>). <code class="func">MOCString</code> returns a string describing the <strong class="pkg">MOC</strong> 3 format of <var class="Arg">moctbl</var>.</p>
<p>If a second argument <var class="Arg">chars</var> is specified, it must be a list of <strong class="pkg">MOC</strong> format characters as returned by <code class="func">MOCChars</code> (<a href="chap6.html#X7BF1B6BF83BDD27E"><span class="RefLink">6.2-6</span></a>). In this case, these characters are stored under label <code class="code">30900</code>. If the second argument is missing then the basic set of ordinary irreducibles is stored under this label.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">moca5:= MOCTable( CharacterTable( "A5" ) );</span>
rec( 30170 := [ [ ], [ 2, 2, 1, 1 ], [ 3, 3, 1, 1 ], [ 4, 5, 1, 1 ] ]
,
30900 := [ [ 1, 1, 1, 1, 0 ], [ 3, -1, 0, 0, -1 ],
[ 3, -1, 0, 1, 1 ], [ 4, 0, 1, -1, 0 ], [ 5, 1, -1, 0, 0 ] ],
GAPtbl := CharacterTable( "A5" ), centralizers := [ 60, 4, 3, 5 ],
cycsubgps := [ 1, 2, 3, 4, 4 ],
fieldbases :=
[ CanonicalBasis( Rationals ), CanonicalBasis( Rationals ),
CanonicalBasis( Rationals ),
Basis( NF(5,[ 1, 4 ]), [ 1, E(5)+E(5)^4 ] ) ], fields := [ ],
galconjinfo := [ 1, 1, 2, 1, 3, 1, 4, 1, 4, 2 ],
identifier := "MOCTable(A5)",
invmap := [ [ 1, 1, 0 ], [ 1, 2, 0 ], [ 1, 3, 0 ],
[ 1, 4, 0, 1, 5, 0 ] ], orders := [ 1, 2, 3, 5 ],
powerinfo :=
[ ,
[ [ 1, 1, 0 ], [ 1, 1, 0 ], [ 1, 3, 0 ],
[ 1, 4, -1, 5, 0, -1, 5, 0 ] ],
[ [ 1, 1, 0 ], [ 1, 2, 0 ], [ 1, 1, 0 ],
[ 1, 4, -1, 5, 0, -1, 5, 0 ] ],,
[ [ 1, 1, 0 ], [ 1, 2, 0 ], [ 1, 3, 0 ], [ 1, 1, 0, 0 ] ] ],
prime := 0, repcycsub := [ 1, 2, 3, 4 ],
tensinfo :=
[ [ 1 ], [ 1 ], [ 1 ],
[ 2, 1, 1, 1, 1, 2, 2, 0, 1, 1, 2, 1, 2, 1, -1, 2, 2, 0 ] ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">str:= MOCString( moca5 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">str{[1..68]};</span>
"y100y105ay110fey130t60edfy140bcdfy150bbbfcabbey160bbcbdbebecy170ccbb"
<span class="GAPprompt">gap></span> <span class="GAPinput">moca5mod3:= MOCTable( CharacterTable( "A5" ) mod 3, [ 1 .. 4 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">MOCString( moca5mod3 ){ [ 1 .. 68 ] };</span>
"y100y105dy110edy130t60efy140bcfy150bbfcabbey160bbcbdbdcy170ccbbdfbby"
</pre></div>
<p><a id="X7F17B82879C8F953" name="X7F17B82879C8F953"></a></p>
<h5>6.2-4 ScanMOC</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ScanMOC</code>( <var class="Arg">list</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a record containing the information encoded in the list <var class="Arg">list</var>. The components of the result are the labels that occur in <var class="Arg">list</var>. If <var class="Arg">list</var> is in <strong class="pkg">MOC</strong> 2 format (10000-format), the names of components are 30000-numbers; if it is in <strong class="pkg">MOC</strong> 3 format the names of components have <code class="code">yABC</code>-format.</p>
<p><a id="X7FBA67C585BEBFE5" name="X7FBA67C585BEBFE5"></a></p>
<h5>6.2-5 GAPChars</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GAPChars</code>( <var class="Arg">tbl</var>, <var class="Arg">mocchars</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">tbl</var> be a character table or a <strong class="pkg">MOC</strong> table record, and <var class="Arg">mocchars</var> be either a list of <strong class="pkg">MOC</strong> format characters (as returned by <code class="func">MOCChars</code> (<a href="chap6.html#X7BF1B6BF83BDD27E"><span class="RefLink">6.2-6</span></a>)) or a list of positive integers such as a record component encoding characters, in a record produced by <code class="func">ScanMOC</code> (<a href="chap6.html#X7F17B82879C8F953"><span class="RefLink">6.2-4</span></a>).</p>
<p><code class="func">GAPChars</code> returns translations of <var class="Arg">mocchars</var> to <strong class="pkg">GAP</strong> character values lists.</p>
<p><a id="X7BF1B6BF83BDD27E" name="X7BF1B6BF83BDD27E"></a></p>
<h5>6.2-6 MOCChars</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MOCChars</code>( <var class="Arg">tbl</var>, <var class="Arg">gapchars</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">tbl</var> be a character table or a <strong class="pkg">MOC</strong> table record, and <var class="Arg">gapchars</var> be a list of (<strong class="pkg">GAP</strong> format) characters. <code class="func">MOCChars</code> returns translations of <var class="Arg">gapchars</var> to <strong class="pkg">MOC</strong> format.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">scan:= ScanMOC( str );</span>
rec( y050 := [ 5, 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 1, 0, 0 ],
y105 := [ 0 ], y110 := [ 5, 4 ], y130 := [ 60, 4, 3, 5 ],
y140 := [ 1, 2, 3, 5 ], y150 := [ 1, 1, 1, 5, 2, 0, 1, 1, 4 ],
y160 := [ 1, 1, 2, 1, 3, 1, 4, 1, 4, 2 ],
y170 := [ 2, 2, 1, 1, 3, 3, 1, 1, 4, 5, 1, 1 ],
y210 := [ 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 0, 1, 1, 2, 1, 2, 1, -1, 2,
2, 0 ], y220 := [ 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 4, 0, 1, 5, 0 ],
y230 := [ 2, 1, 1, 0, 1, 1, 0, 1, 3, 0, 1, 4, -1, 5, 0, -1, 5, 0 ],
y900 := [ 1, 1, 1, 1, 0, 3, -1, 0, 0, -1, 3, -1, 0, 1, 1, 4, 0, 1,
-1, 0, 5, 1, -1, 0, 0 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">gapchars:= GAPChars( moca5, scan.y900 );</span>
[ [ 1, 1, 1, 1, 1 ], [ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ],
[ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ], [ 4, 0, 1, -1, -1 ],
[ 5, 1, -1, 0, 0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">mocchars:= MOCChars( moca5, gapchars );</span>
[ [ 1, 1, 1, 1, 0 ], [ 3, -1, 0, 0, -1 ], [ 3, -1, 0, 1, 1 ],
[ 4, 0, 1, -1, 0 ], [ 5, 1, -1, 0, 0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Concatenation( mocchars ) = scan.y900;</span>
true
</pre></div>
<p><a id="X85BBAB537B4B72F5" name="X85BBAB537B4B72F5"></a></p>
<h4>6.3 <span class="Heading">Interface to <strong class="pkg">GAP</strong> 3</span></h4>
<p>The following functions are used to read and write character tables in <strong class="pkg">GAP</strong> 3 format.</p>
<p><a id="X7DB321DE80046931" name="X7DB321DE80046931"></a></p>
<h5>6.3-1 GAP3CharacterTableScan</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GAP3CharacterTableScan</code>( <var class="Arg">string</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">string</var> be a string that contains the output of the <strong class="pkg">GAP</strong> 3 function <code class="code">PrintCharTable</code>. In other words, <var class="Arg">string</var> describes a <strong class="pkg">GAP</strong> record whose components define an ordinary character table object in <strong class="pkg">GAP</strong> 3. <code class="func">GAP3CharacterTableScan</code> returns the corresponding <strong class="pkg">GAP</strong> 4 character table object.</p>
<p>The supported record components are given by the list <code class="func">GAP3CharacterTableData</code> (<a href="chap6.html#X7B2725D2789A9D85"><span class="RefLink">6.3-3</span></a>).</p>
<p><a id="X7B2C53137D1227FE" name="X7B2C53137D1227FE"></a></p>
<h5>6.3-2 GAP3CharacterTableString</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GAP3CharacterTableString</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For an ordinary character table <var class="Arg">tbl</var>, <code class="func">GAP3CharacterTableString</code> returns a string that when read into <strong class="pkg">GAP</strong> 3 evaluates to a character table corresponding to <var class="Arg">tbl</var>. A similar format is printed by the <strong class="pkg">GAP</strong> 3 function <code class="code">PrintCharTable</code>.</p>
<p>The supported record components are given by the list <code class="func">GAP3CharacterTableData</code> (<a href="chap6.html#X7B2725D2789A9D85"><span class="RefLink">6.3-3</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "Alternating", 5 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">str:= GAP3CharacterTableString( tbl );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Print( str );</span>
rec(
centralizers := [ 60, 4, 3, 5, 5 ],
fusions := [ rec( map := [ 1, 3, 4, 7, 7 ], name := "Sym(5)" ) ],
identifier := "Alt(5)",
irreducibles := [
[ 1, 1, 1, 1, 1 ],
[ 4, 0, 1, -1, -1 ],
[ 5, 1, -1, 0, 0 ],
[ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ],
[ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ]
],
orders := [ 1, 2, 3, 5, 5 ],
powermap := [ , [ 1, 1, 3, 5, 4 ], [ 1, 2, 1, 5, 4 ], , [ 1, 2, 3, 1, \
1 ] ],
size := 60,
text := "computed using generic character table for alternating groups\
",
operations := CharTableOps )
<span class="GAPprompt">gap></span> <span class="GAPinput">scan:= GAP3CharacterTableScan( str );</span>
CharacterTable( "Alt(5)" )
<span class="GAPprompt">gap></span> <span class="GAPinput">TransformingPermutationsCharacterTables( tbl, scan );</span>
rec( columns := (), group := Group([ (4,5) ]), rows := () )
</pre></div>
<p><a id="X7B2725D2789A9D85" name="X7B2725D2789A9D85"></a></p>
<h5>6.3-3 GAP3CharacterTableData</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GAP3CharacterTableData</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>This is a list of pairs, the first entry being the name of a component in a <strong class="pkg">GAP</strong> 3 character table and the second entry being the corresponding attribute name in <strong class="pkg">GAP</strong> 4. The variable is used by <code class="func">GAP3CharacterTableScan</code> (<a href="chap6.html#X7DB321DE80046931"><span class="RefLink">6.3-1</span></a>) and <code class="func">GAP3CharacterTableString</code> (<a href="chap6.html#X7B2C53137D1227FE"><span class="RefLink">6.3-2</span></a>).</p>
<p><a id="X7806073380076800" name="X7806073380076800"></a></p>
<h4>6.4 <span class="Heading">Interface to the Cambridge Format</span></h4>
<p>The following functions deal with the so-called Cambridge format, in which the source data of the character tables in the <strong class="pkg">Atlas</strong> of Finite Groups <a href="chapBib.html#biBCCN85">[CCNPW85]</a> and in the <strong class="pkg">Atlas</strong> of Brauer Characters <a href="chapBib.html#biBJLPW95">[JLPW95]</a> are stored. Each such table is stored on a file of its own. The line length is at most <span class="SimpleMath">78</span>, and each item of the table starts in a new line, behind one of the following prefixes.</p>
<dl>
<dt><strong class="Mark"><code class="code">#23</code></strong></dt>
<dd><p>a description and the name(s) of the simple group</p>
</dd>
<dt><strong class="Mark"><code class="code">#7</code></strong></dt>
<dd><p>integers describing the column widths</p>
</dd>
<dt><strong class="Mark"><code class="code">#9</code></strong></dt>
<dd><p>the symbols <code class="code">;</code> and <code class="code">@</code>, denoting columns between tables and columns that belong to conjugacy classes, respectively</p>
</dd>
<dt><strong class="Mark"><code class="code">#1</code></strong></dt>
<dd><p>the symbol <code class="code">|</code> in columns between tables, and centralizer orders otherwise</p>
</dd>
<dt><strong class="Mark"><code class="code">#2</code></strong></dt>
<dd><p>the symbols <code class="code">p</code> (in the first column only), <code class="code">power</code> (in the second column only, which belongs to the class of the identity element), <code class="code">|</code> in other columns between tables, and descriptions of the powers of classes otherwise</p>
</dd>
<dt><strong class="Mark"><code class="code">#3</code></strong></dt>
<dd><p>the symbols <code class="code">p'</code> (in the first column only), <code class="code">part</code> (in the second column only, which belongs to the class of the identity element), <code class="code">|</code> in other columns between tables, and descriptions of the <span class="SimpleMath">p</span>-prime parts of classes otherwise</p>
</dd>
<dt><strong class="Mark"><code class="code">#4</code></strong></dt>
<dd><p>the symbols <code class="code">ind</code> and <code class="code">fus</code> in columns between tables, and class names otherwise</p>
</dd>
<dt><strong class="Mark"><code class="code">#5</code></strong></dt>
<dd><p>either <code class="code">|</code> or strings composed from the symbols <code class="code">+</code>, <code class="code">-</code>, <code class="code">o</code>, and integers in columns where the lines starting with <code class="code">#4</code> contain <code class="code">ind</code>; the symbols <code class="code">:</code>, <code class="code">.</code>, <code class="code">?</code> in columns where these lines contain <code class="code">fus</code>; character values or <code class="code">|</code> otherwise</p>
</dd>
<dt><strong class="Mark"><code class="code">#6</code></strong></dt>
<dd><p>the symbols <code class="code">|</code>, <code class="code">ind</code>, <code class="code">and</code>, and <code class="code">fus</code> in columns between tables; the symbol <code class="code">|</code> and element orders of preimage classes in downward extensions otherwise</p>
</dd>
<dt><strong class="Mark"><code class="code">#8</code></strong></dt>
<dd><p>the last line of the data, may contain the date of the last change</p>
</dd>
<dt><strong class="Mark"><code class="code">#C</code></strong></dt>
<dd><p>comments.</p>
</dd>
</dl>
<p><a id="X78A84889878D98BB" name="X78A84889878D98BB"></a></p>
<h5>6.4-1 CambridgeMaps</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CambridgeMaps</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a character table <var class="Arg">tbl</var>, <code class="func">CambridgeMaps</code> returns a record with the following components.</p>
<dl>
<dt><strong class="Mark"><code class="code">name</code></strong></dt>
<dd><p>a list of strings denoting class names,</p>
</dd>
<dt><strong class="Mark"><code class="code">power</code></strong></dt>
<dd><p>a list of strings, the <span class="SimpleMath">i</span>-th entry encodes the <span class="SimpleMath">p</span>-th powers of the <span class="SimpleMath">i</span>-th class, for all prime divisors <span class="SimpleMath">p</span> of the group order,</p>
</dd>
<dt><strong class="Mark"><code class="code">prime</code></strong></dt>
<dd><p>a list of strings, the <span class="SimpleMath">i</span>-th entry encodes the <span class="SimpleMath">p</span>-prime parts of the <span class="SimpleMath">i</span>-th class, for all prime divisors <span class="SimpleMath">p</span> of the group order.</p>
</dd>
</dl>
<p>The meaning of the entries of the lists is defined in <a href="chapBib.html#biBCCN85">[CCNPW85, Chapter 7, Sections 3–5]</a>).</p>
<p><code class="func">CambridgeMaps</code> is used for example by <code class="func">Display</code> (<span class="RefLink">Reference: Display (for a character table)</span>) in the case that the <code class="code">powermap</code> option has the value <code class="code">"ATLAS"</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CambridgeMaps( CharacterTable( "A5" ) );</span>
rec( names := [ "1A", "2A", "3A", "5A", "B*" ],
power := [ "", "A", "A", "A", "A" ],
prime := [ "", "A", "A", "A", "A" ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">CambridgeMaps( CharacterTable( "A5" ) mod 2 );</span>
rec( names := [ "1A", "3A", "5A", "B*" ],
power := [ "", "A", "A", "A" ], prime := [ "", "A", "A", "A" ] )
</pre></div>
<p><a id="X8209063C813240E5" name="X8209063C813240E5"></a></p>
<h5>6.4-2 StringOfCambridgeFormat</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StringOfCambridgeFormat</code>( <var class="Arg">tbls</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><em>(This function is experimental.)</em></p>
<p>Let <var class="Arg">tbls</var> be a list of character tables, which are central extensions of the first entry in <var class="Arg">tbls</var>, and such that the factor fusion to the first entry is stored on all other tables in the list.</p>
<p><code class="func">StringOfCambridgeFormat</code> returns a string that encodes an approximation of the Cambridge format file for the first entry in <var class="Arg">tbls</var>. Differences to the original format may occur for irrational character values; the descriptions of these values have been chosen deliberately for the original files, it is not obvious how to compute these descriptions from the character tables in question.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:= CharacterTable( "A5" );; 2t:= CharacterTable( "2.A5" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Print( StringOfCambridgeFormat( [ t, 2t ] ) );</span>
#23 ? A5
#7 4 4 4 4 4 4
#9 ; @ @ @ @ @
#1 | 60 4 3 5 5
#2 p power A A A A
#3 p' part A A A A
#4 ind 1A 2A 3A 5A B*
#5 + 1 1 1 1 1
#5 + 3 -1 0 -b5 *
#5 + 3 -1 0 * -b5
#5 + 4 0 1 -1 -1
#5 + 5 1 -1 0 0
#6 ind 1 4 3 5 5
#6 | 2 | 6 10 10
#5 - 2 0 -1 b5 *
#5 - 2 0 -1 * b5
#5 - 4 0 1 -1 -1
#5 - 6 0 0 1 1
#8
</pre></div>
<p><a id="X863CB6737FF9B740" name="X863CB6737FF9B740"></a></p>
<h4>6.5 <span class="Heading">Interface to the <strong class="pkg">MAGMA</strong> System</span></h4>
<p>This interface is intended to convert character tables given in <strong class="pkg">MAGMA</strong>'s display format into <strong class="pkg">GAP</strong> character tables.</p>
<p>The function <code class="func">BosmaBase</code> (<a href="chap6.html#X79D160BD7ECA6D2F"><span class="RefLink">6.5-1</span></a>) is used for the translation of irrational values; this function may be of interest independent of the conversion of character tables.</p>
<p><a id="X79D160BD7ECA6D2F" name="X79D160BD7ECA6D2F"></a></p>
<h5>6.5-1 BosmaBase</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BosmaBase</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a positive integer <var class="Arg">n</var> that is not congruent to <span class="SimpleMath">2</span> modulo <span class="SimpleMath">4</span>, <code class="func">BosmaBase</code> returns the list of exponents <span class="SimpleMath">i</span> for which <code class="code">E(<var class="Arg">n</var>)^</code><span class="SimpleMath">i</span> belongs to the canonical basis of the <var class="Arg">n</var>-th cyclotomic field that is defined in <a href="chapBib.html#biBBos90">[Bos90, Section 5]</a>.</p>
<p>As a set, this basis is defined as follows. Let <span class="SimpleMath">P</span> denote the set of prime divisors of <var class="Arg">n</var> and <var class="Arg">n</var> <span class="SimpleMath">= ∏_{p ∈ P} n_p</span>. Let <span class="SimpleMath">e_l =</span> <code class="code">E</code><span class="SimpleMath">(l)</span> for any positive integer <span class="SimpleMath">l</span>, and <span class="SimpleMath">{ e_{m_1}^j }_{j ∈ J} ⊗ { e_{m_2}^k }_{k ∈ K} = { e_{m_1}^j ⋅ e_{m_2}^k }_{j ∈ J, k ∈ K}</span> for any positive integers <span class="SimpleMath">m_1</span>, <span class="SimpleMath">m_2</span>. (This notation is the same as the one used in the description of <code class="func">ZumbroichBase</code> (<span class="RefLink">Reference: ZumbroichBase</span>).)</p>
<p>Then the basis is</p>
<p class="pcenter">B_n = ⨂_{p ∈ P} B_{n_p}</p>
<p>where</p>
<p class="pcenter">B_{n_p} = { e_{n_p}^k; 0 ≤ k ≤ φ(n_p)-1 };</p>
<p>here <span class="SimpleMath">φ</span> denotes Euler's function, see <code class="func">Phi</code> (<span class="RefLink">Reference: Phi</span>).</p>
<p><span class="SimpleMath">B_n</span> consists of roots of unity, it is an integral basis (that is, exactly the integral elements in <span class="SimpleMath">ℚ_n</span> have integral coefficients w.r.t. <span class="SimpleMath">B_n</span>, cf. <code class="func">IsIntegralCyclotomic</code> (<span class="RefLink">Reference: IsIntegralCyclotomic</span>)), and for any divisor <span class="SimpleMath">m</span> of <var class="Arg">n</var> that is not congruent to <span class="SimpleMath">2</span> modulo <span class="SimpleMath">4</span>, <span class="SimpleMath">B_m</span> is a subset of <span class="SimpleMath">B_n</span>.</p>
<p>Note that the list <span class="SimpleMath">l</span>, say, that is returned by <code class="func">BosmaBase</code> is in general not a set. The ordering of the elements in <span class="SimpleMath">l</span> fits to the coefficient lists for irrational values used by <strong class="pkg">MAGMA</strong>'s display format.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">b:= BosmaBase( 8 );</span>
[ 0, 1, 2, 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">b:= Basis( CF(8), List( b, i -> E(8)^i ) );</span>
Basis( CF(8), [ 1, E(8), E(4), E(8)^3 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">Coefficients( b, Sqrt(2) );</span>
[ 0, 1, 0, -1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Coefficients( b, Sqrt(-2) );</span>
[ 0, 1, 0, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">b:= BosmaBase( 15 );</span>
[ 0, 5, 3, 8, 6, 11, 9, 14 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">b:= List( b, i -> E(15)^i );</span>
[ 1, E(3), E(5), E(15)^8, E(5)^2, E(15)^11, E(5)^3, E(15)^14 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Coefficients( Basis( CF(15), b ), EB(15) );</span>
[ -1, -1, 0, 0, -1, -2, -1, -2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">BosmaBase( 48 );</span>
[ 0, 3, 6, 9, 12, 15, 18, 21, 16, 19, 22, 25, 28, 31, 34, 37 ]
</pre></div>
<p><a id="X8690C41E7F8EEDF9" name="X8690C41E7F8EEDF9"></a></p>
<h5>6.5-2 GAPTableOfMagmaFile</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GAPTableOfMagmaFile</code>( <var class="Arg">file</var>, <var class="Arg">identifier</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">file</var> be the name of a file that contains a character table in <strong class="pkg">MAGMA</strong>'s display format, and <var class="Arg">identifier</var> be a string. <code class="func">GAPTableOfMagmaFile</code> returns the corresponding <strong class="pkg">GAP</strong> character table.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tmpdir:= DirectoryTemporary();;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">file:= Filename( tmpdir, "magmatable" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">str:= "\</span>
<span class="GAPprompt">></span> <span class="GAPinput">Character Table of Group G\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">--------------------------\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">---------------------------\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">Class | 1 2 3 4 5\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">Size | 1 15 20 12 12\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">Order | 1 2 3 5 5\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">---------------------------\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">p = 2 1 1 3 5 4\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">p = 3 1 2 1 5 4\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">p = 5 1 2 3 1 1\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">---------------------------\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">X.1 + 1 1 1 1 1\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">X.2 + 3 -1 0 Z1 Z1#2\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">X.3 + 3 -1 0 Z1#2 Z1\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">X.4 + 4 0 1 -1 -1\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">X.5 + 5 1 -1 0 0\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">Explanation of Character Value Symbols\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">--------------------------------------\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput"># denotes algebraic conjugation, that is,\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">#k indicates replacing the root of unity w by w^k\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">Z1 = (CyclotomicField(5: Sparse := true)) ! [\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">RationalField() | 1, 0, 1, 1 ]\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">FileString( file, str );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= GAPTableOfMagmaFile( file, "MagmaA5" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( tbl );</span>
MagmaA5
2 2 2 . . .
3 1 . 1 . .
5 1 . . 1 1
1a 2a 3a 5a 5b
2P 1a 1a 3a 5b 5a
3P 1a 2a 1a 5b 5a
5P 1a 2a 3a 1a 1a
X.1 1 1 1 1 1
X.2 3 -1 . A *A
X.3 3 -1 . *A A
X.4 4 . 1 -1 -1
X.5 5 1 -1 . .
A = -E(5)-E(5)^4
= (1-Sqrt(5))/2 = -b5
<span class="GAPprompt">gap></span> <span class="GAPinput">str:= "\</span>
<span class="GAPprompt">></span> <span class="GAPinput">Character Table of Group G\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">--------------------------\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">------------------------------\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">Class | 1 2 3 4 5 6\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">Size | 1 1 1 1 1 1\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">Order | 1 2 3 3 6 6\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">------------------------------\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">p = 2 1 1 4 3 3 4\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">p = 3 1 2 1 1 2 2\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">------------------------------\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">X.1 + 1 1 1 1 1 1\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">X.2 + 1 -1 1 1 -1 -1\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">X.3 0 1 1 J-1-J-1-J J\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">X.4 0 1 -1 J-1-J 1+J -J\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">X.5 0 1 1-1-J J J-1-J\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">X.6 0 1 -1-1-J J -J 1+J\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">Explanation of Character Value Symbols\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">--------------------------------------\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">J = RootOfUnity(3)\n\</span>
<span class="GAPprompt">></span> <span class="GAPinput">";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">FileString( file, str );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= GAPTableOfMagmaFile( file, "MagmaC6" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( tbl );</span>
MagmaC6
2 1 1 1 1 1 1
3 1 1 1 1 1 1
1a 2a 3a 3b 6a 6b
2P 1a 1a 3b 3a 3a 3b
3P 1a 2a 1a 1a 2a 2a
X.1 1 1 1 1 1 1
X.2 1 -1 1 1 -1 -1
X.3 1 1 A /A /A A
X.4 1 -1 A /A -/A -A
X.5 1 1 /A A A /A
X.6 1 -1 /A A -A -/A
A = E(3)
= (-1+Sqrt(-3))/2 = b3
</pre></div>
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