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<a id="X81C250407B91443F" name="X81C250407B91443F"></a>
<div class="ChapSects"><a href="chap6.html#X81C250407B91443F">6 Interfaces to Other Data Formats for Character Tables</a>
<div class="ContSect"> <a href="chap6.html#X79F5E5A283E0D190">6.1 Interface to the CAS System</a>

<div class="ContSSBlock">
 <a href="chap6.html#X841537DA7E495B73">6.1-1 CASString</a>
</div></div>
<div class="ContSect"> <a href="chap6.html#X835EE8028539BB63">6.2 Interface to the MOC System</a>

<div class="ContSSBlock">
 <a href="chap6.html#X842957217E3697A1">6.2-1 MAKElb11</a>
 <a href="chap6.html#X832005B37F0A2872">6.2-2 MOCTable</a>
 <a href="chap6.html#X7FBC528C7818814C">6.2-3 MOCString</a>
 <a href="chap6.html#X7F17B82879C8F953">6.2-4 ScanMOC</a>
 <a href="chap6.html#X7FBA67C585BEBFE5">6.2-5 GAPChars</a>
 <a href="chap6.html#X7BF1B6BF83BDD27E">6.2-6 MOCChars</a>
</div></div>
<div class="ContSect"> <a href="chap6.html#X80E1850F7DCED030">6.3 Interface to GAP 3</a>

<div class="ContSSBlock">
 <a href="chap6.html#X7DB321DE80046931">6.3-1 GAP3CharacterTableScan</a>
 <a href="chap6.html#X7B2C53137D1227FE">6.3-2 GAP3CharacterTableString</a>
 <a href="chap6.html#X7B2725D2789A9D85">6.3-3 GAP3CharacterTableData</a>
</div></div>
<div class="ContSect"> <a href="chap6.html#X7806073380076800">6.4 Interface to the Cambridge Format</a>

<div class="ContSSBlock">
 <a href="chap6.html#X78A84889878D98BB">6.4-1 CambridgeMaps</a>
 <a href="chap6.html#X8209063C813240E5">6.4-2 StringOfCambridgeFormat</a>
</div></div>
<div class="ContSect"> <a href="chap6.html#X8676916F81F40A0F">6.5 Interface to the MAGMA System</a>

<div class="ContSSBlock">
 <a href="chap6.html#X79D160BD7ECA6D2F">6.5-1 BosmaBase</a>
 <a href="chap6.html#X7B1FAE1A7FD5D5A1">6.5-2 GAPTableOfMagmaFile</a>
 <a href="chap6.html#X824D2B4A79A9E5AE">6.5-3 CharacterTableComputedByMagma</a>
</div></div>
</div>

<h3>6 Interfaces to Other Data Formats for Character Tables</h3>

This chapter describes data formats for character tables that can be read or created by GAP. Currently these are the formats used by

<ul>
<li>the CAS system (see <a href="chap6.html#X79F5E5A283E0D190">6.1</a>),

</li>
<li>the MOC system (see <a href="chap6.html#X835EE8028539BB63">6.2</a>),

</li>
<li>GAP 3 (see <a href="chap6.html#X80E1850F7DCED030">6.3</a>),

</li>
<li>the so-called Cambridge format (see <a href="chap6.html#X7806073380076800">6.4</a>), and

</li>
<li>the MAGMA system (see <a href="chap6.html#X8676916F81F40A0F">6.5</a>).

</li>
</ul>
<a id="X79F5E5A283E0D190" name="X79F5E5A283E0D190"></a>

<h4>6.1 Interface to the CAS System</h4>

The interface to CAS (see <a href="chapBib.html#biBNPP84">[NPP84]</a>) is thought just for printing the CAS data to a file. The function <code class="func">CASString</code> (<a href="chap6.html#X841537DA7E495B73">6.1-1</a>) is available mainly in order to document the data format. Reading CAS tables is not supported; note that the tables contained in the CAS Character Table Library have been migrated to GAP using a few <code class="code">sed</code> scripts and <code class="code">C</code> programs.

<a id="X841537DA7E495B73" name="X841537DA7E495B73"></a>

<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>
is a string that encodes the CAS 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 CAS system using its <code class="code">get</code> command (see <a href="chapBib.html#biBNPP84">[NPP84]</a>).

The used line length is the first entry in the list returned by <code class="func">SizeScreen</code> (<a href="../../../doc/ref/chap6.html#X8723E0A1837894F3">Reference: SizeScreen</a>).

Only the known values of the following attributes are used. <code class="func">ClassParameters</code> (<a href="../../../doc/ref/chap71.html#X8333E8038308947E">Reference: ClassParameters</a>) (for partitions only), <code class="func">ComputedClassFusions</code> (<a href="../../../doc/ref/chap73.html#X7F71402285B7DE8E">Reference: ComputedClassFusions</a>), <code class="func">ComputedIndicators</code> (<a href="../../../doc/ref/chap71.html#X7FD3D3047DE6381E">Reference: ComputedIndicators</a>), <code class="func">ComputedPowerMaps</code> (<a href="../../../doc/ref/chap73.html#X781FAA497E3B4D1A">Reference: ComputedPowerMaps</a>), <code class="func">ComputedPrimeBlocks</code> (<a href="../../../doc/ref/chap71.html#X7ACB9306804F4E3F">Reference: ComputedPrimeBlockss</a>), <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F">Reference: Identifier for character tables</a>), <code class="func">InfoText</code> (<a href="../../../doc/ref/chap12.html#X871562FD7F982C12">Reference: InfoText</a>), <code class="func">Irr</code> (<a href="../../../doc/ref/chap71.html#X873B3CC57E9A5492">Reference: Irr</a>), <code class="func">OrdersClassRepresentatives</code> (<a href="../../../doc/ref/chap71.html#X86F455DA7A9C30EE">Reference: OrdersClassRepresentatives</a>), <code class="func">Size</code> (<a href="../../../doc/ref/chap30.html#X858ADA3B7A684421">Reference: Size</a>), <code class="func">SizesCentralizers</code> (<a href="../../../doc/ref/chap71.html#X7CF7907F790A5DE6">Reference: SizesCentralisers</a>).

<div class="example"><pre>
gap> Print( CASString( CharacterTable( "Cyclic", 2 ) ), "\n" );
'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>

<a id="X835EE8028539BB63" name="X835EE8028539BB63"></a>

<h4>6.2 Interface to the MOC System</h4>

The interface to MOC (see <a href="chapBib.html#biBHJLP92">[HJLP]</a>) can be used to print MOC input. Additionally it provides an alternative representation of (virtual) characters.

The MOC 3 code of a 5 digit number in MOC 2 code is given by the following list. (Note that the code must contain only lower case letters.)

<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>

Note that any long number in MOC 2 format is divided into packages of length 4, the first (!) one filled with leading zeros if necessary. Such a number with decimals d_1, d_2, ..., d_{4n+k} is the sequence 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} where 0 ≤ k ≤ 3, the first digit of x is 1 if the number is positive and 2 if the number is negative, and then follow (4-k) zeros.

Details about the MOC 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>.

<a id="X842957217E3697A1" name="X842957217E3697A1"></a>

<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>
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.

The output of <code class="func">MAKElb11</code> is used by the MOC system; Calling <code class="code">MAKElb11( [ 3 .. 189 ] )</code> will print something very similar to Richard Parker's file lb11.

<div class="example"><pre>
gap> MAKElb11( [ 3, 4 ] );
 3 2 0 1 0
 4 2 0 1 0
</pre></div>

<a id="X832005B37F0A2872" name="X832005B37F0A2872"></a>

<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>
<code class="func">MOCTable</code> returns the MOC table record of the GAP character table <var class="Arg">gaptbl</var>.

The one argument version can be used only if <var class="Arg">gaptbl</var> is an ordinary (G.0) table. For Brauer (G.p) 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> (<a href="../../../doc/ref/chap71.html#X873B3CC57E9A5492">Reference: Irr</a>) list of the ordinary table of <var class="Arg">gaptbl</var>.

The result is a record that contains the information of <var class="Arg">gaptbl</var> in a format similar to the MOC 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">6.2-3</a>).

The components of the result are

<dl>
<dt><code class="code">identifier</code></dt>
<dd>the string <code class="code">MOCTable( </code>name<code class="code"> )</code> where name is the <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F">Reference: Identifier for character tables</a>) value of <var class="Arg">gaptbl</var>,

</dd>
<dt><code class="code">GAPtbl</code></dt>
<dd><var class="Arg">gaptbl</var>,

</dd>
<dt><code class="code">prime</code></dt>
<dd>the characteristic of the field (label <code class="code">30105</code> in MOC),

</dd>
<dt><code class="code">centralizers</code></dt>
<dd>centralizer orders for cyclic subgroups (label <code class="code">30130</code>)

</dd>
<dt><code class="code">orders</code></dt>
<dd>element orders for cyclic subgroups (label <code class="code">30140</code>)

</dd>
<dt><code class="code">fieldbases</code></dt>
<dd>at position i the Parker basis of the number field generated by the character values of the i-th cyclic subgroup. The length of <code class="code">fieldbases</code> is equal to the value of label <code class="code">30110</code> in MOC.

</dd>
<dt><code class="code">cycsubgps</code></dt>
<dd><code class="code">cycsubgps[i] = j</code> means that class <code class="code">i</code> of the GAP table belongs to the <code class="code">j</code>-th cyclic subgroup of the GAP table,

</dd>
<dt><code class="code">repcycsub</code></dt>
<dd><code class="code">repcycsub[j] = i</code> means that class <code class="code">i</code> of the GAP table is the representative of the <code class="code">j</code>-th cyclic subgroup of the GAP table. Note that the representatives of GAP table and MOC table need not agree!

</dd>
<dt><code class="code">galconjinfo</code></dt>
<dd>a list [ r_1, c_1, r_2, c_2, ..., r_n, c_n ] which means that the i-th class of the GAP table is the c_i-th conjugate of the representative of the r_i-th cyclic subgroup on the MOC table. (This is used to translate back to GAP format, stored under label <code class="code">30160</code>)

</dd>
<dt><code class="code">30170</code></dt>
<dd>(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 GAP.

</dd>
<dt><code class="code">tensinfo</code></dt>
<dd>tensor product information, used to compute the coefficients of the Parker base for tensor products of characters (label <code class="code">30210</code> in MOC). For a field with vector space basis (v_1, v_2, ..., v_n), the tensor product information of a cyclic subgroup in MOC (as computed by <code class="code">fct</code>) is either 1 (for rational classes) or a sequence

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

which means that the coefficient of v_k in the product

( ∑_i=1^n a_i v_i ) ( ∑_j=1^n b_j v_j )

is equal to

∑_i=1^m_k x_k,i a_y_k,i} b_z_k,i} .

On a MOC table in GAP, the <code class="code">tensinfo</code> component is a list of lists, each containing exactly the sequence mentioned above.

</dd>
<dt><code class="code">invmap</code></dt>
<dd>inverse map to compute complex conjugate characters, label <code class="code">30220</code> in MOC.

</dd>
<dt><code class="code">powerinfo</code></dt>
<dd>field embeddings for p-th symmetrizations, p a prime integer not larger than the largest element order, label <code class="code">30230</code> in MOC.

</dd>
<dt><code class="code">30900</code></dt>
<dd>basic set of restricted ordinary irreducibles in the case of nonzero characteristic, all ordinary irreducibles otherwise.

</dd>
</dl>
<a id="X7FBC528C7818814C" name="X7FBC528C7818814C"></a>

<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>
Let <var class="Arg">moctbl</var> be a MOC table record, as returned by <code class="func">MOCTable</code> (<a href="chap6.html#X832005B37F0A2872">6.2-2</a>). <code class="func">MOCString</code> returns a string describing the MOC 3 format of <var class="Arg">moctbl</var>.

If a second argument <var class="Arg">chars</var> is specified, it must be a list of MOC format characters as returned by <code class="func">MOCChars</code> (<a href="chap6.html#X7BF1B6BF83BDD27E">6.2-6</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.

<div class="example"><pre>
gap> moca5:= MOCTable( CharacterTable( "A5" ) );
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 ] ] )
gap> str:= MOCString( moca5 );;
gap> str{[1..68]};
"y100y105ay110fey130t60edfy140bcdfy150bbbfcabbey160bbcbdbebecy170ccbb"
gap> moca5mod3:= MOCTable( CharacterTable( "A5" ) mod 3, [ 1 .. 4 ] );;
gap> MOCString( moca5mod3 ){ [ 1 .. 68 ] };
"y100y105dy110edy130t60efy140bcfy150bbfcabbey160bbcbdbdcy170ccbbdfbby"
</pre></div>

<a id="X7F17B82879C8F953" name="X7F17B82879C8F953"></a>

<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>
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 MOC 2 format (10000-format), the names of components are 30000-numbers; if it is in MOC 3 format the names of components have <code class="code">yABC</code>-format.

<a id="X7FBA67C585BEBFE5" name="X7FBA67C585BEBFE5"></a>

<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>
Let <var class="Arg">tbl</var> be a character table or a MOC table record, and <var class="Arg">mocchars</var> be either a list of MOC format characters (as returned by <code class="func">MOCChars</code> (<a href="chap6.html#X7BF1B6BF83BDD27E">6.2-6</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">6.2-4</a>).

<code class="func">GAPChars</code> returns translations of <var class="Arg">mocchars</var> to GAP character values lists.

<a id="X7BF1B6BF83BDD27E" name="X7BF1B6BF83BDD27E"></a>

<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>
Let <var class="Arg">tbl</var> be a character table or a MOC table record, and <var class="Arg">gapchars</var> be a list of (GAP format) characters. <code class="func">MOCChars</code> returns translations of <var class="Arg">gapchars</var> to MOC format.

<div class="example"><pre>
gap> scan:= ScanMOC( str );
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 ] )
gap> gapchars:= GAPChars( moca5, scan.y900 );
[ [ 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 ] ]
gap> mocchars:= MOCChars( moca5, gapchars );
[ [ 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 ] ]
gap> Concatenation( mocchars ) = scan.y900;
true
</pre></div>

<a id="X80E1850F7DCED030" name="X80E1850F7DCED030"></a>

<h4>6.3 Interface to GAP 3</h4>

The following functions are used to read and write character tables in GAP 3 format.

<a id="X7DB321DE80046931" name="X7DB321DE80046931"></a>

<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>
Let <var class="Arg">string</var> be a string that contains the output of the GAP 3 function <code class="code">PrintCharTable</code>. In other words, <var class="Arg">string</var> describes a GAP record whose components define an ordinary character table object in GAP 3. <code class="func">GAP3CharacterTableScan</code> returns the corresponding GAP 4 character table object.

The supported record components are given by the list <code class="func">GAP3CharacterTableData</code> (<a href="chap6.html#X7B2725D2789A9D85">6.3-3</a>).

<a id="X7B2C53137D1227FE" name="X7B2C53137D1227FE"></a>

<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>
For an ordinary character table <var class="Arg">tbl</var>, <code class="func">GAP3CharacterTableString</code> returns a string that when read into GAP 3 evaluates to a character table corresponding to <var class="Arg">tbl</var>. A similar format is printed by the GAP 3 function <code class="code">PrintCharTable</code>.

The supported record components are given by the list <code class="func">GAP3CharacterTableData</code> (<a href="chap6.html#X7B2725D2789A9D85">6.3-3</a>).

<div class="example"><pre>
gap> tbl:= CharacterTable( "Alternating", 5 );;
gap> str:= GAP3CharacterTableString( tbl );;
gap> Print( str );
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 )
gap> scan:= GAP3CharacterTableScan( str );
CharacterTable( "Alt(5)" )
gap> TransformingPermutationsCharacterTables( tbl, scan );
rec( columns := (), group := Group([ (4,5) ]), rows := () )
</pre></div>

<a id="X7B2725D2789A9D85" name="X7B2725D2789A9D85"></a>

<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>
This is a list of pairs, the first entry being the name of a component in a GAP 3 character table and the second entry being the corresponding attribute name in GAP 4. The variable is used by <code class="func">GAP3CharacterTableScan</code> (<a href="chap6.html#X7DB321DE80046931">6.3-1</a>) and <code class="func">GAP3CharacterTableString</code> (<a href="chap6.html#X7B2C53137D1227FE">6.3-2</a>).

<a id="X7806073380076800" name="X7806073380076800"></a>

<h4>6.4 Interface to the Cambridge Format</h4>

The following functions deal with the so-called Cambridge format, in which the source data of the character tables in the Atlas of Finite Groups <a href="chapBib.html#biBCCN85">[CCN+85]</a> and in the Atlas 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 78, and each item of the table starts in a new line, behind one of the following prefixes.

<dl>
<dt><code class="code">#23</code></dt>
<dd>a description and the name(s) of the simple group

</dd>
<dt><code class="code">#7</code></dt>
<dd>integers describing the column widths

</dd>
<dt><code class="code">#9</code></dt>
<dd>the symbols <code class="code">;</code> and <code class="code">@</code>, denoting columns between tables and columns that belong to conjugacy classes, respectively

</dd>
<dt><code class="code">#1</code></dt>
<dd>the symbol <code class="code">|</code> in columns between tables, and centralizer orders otherwise

</dd>
<dt><code class="code">#2</code></dt>
<dd>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

</dd>
<dt><code class="code">#3</code></dt>
<dd>the symbols <code class="code">p' (in the first column only), part (in the second column only, which belongs to the class of the identity element), | in other columns between tables, and descriptions of the p-prime parts of classes otherwise

</dd>
<dt><code class="code">#4</code></dt>
<dd>the symbols <code class="code">ind</code> and <code class="code">fus</code> in columns between tables, and class names otherwise

</dd>
<dt><code class="code">#5</code></dt>
<dd>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

</dd>
<dt><code class="code">#6</code></dt>
<dd>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

</dd>
<dt><code class="code">#8</code></dt>
<dd>the last line of the data, may contain the date of the last change

</dd>
<dt><code class="code">#C</code></dt>
<dd>comments.

</dd>
</dl>
<a id="X78A84889878D98BB" name="X78A84889878D98BB"></a>

<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>
For a character table <var class="Arg">tbl</var>, <code class="func">CambridgeMaps</code> returns a record with the following components.

<dl>
<dt><code class="code">names</code></dt>
<dd>a list of strings denoting class names,

</dd>
<dt><code class="code">power</code></dt>
<dd>a list of strings, the i-th entry encodes the p-th powers of the i-th class, for all prime divisors p of the group order,

</dd>
<dt><code class="code">prime</code></dt>
<dd>a list of strings, the i-th entry encodes the p-prime parts of the i-th class, for all prime divisors p of the group order.

</dd>
</dl>
The meaning of the entries of the lists is defined in <a href="chapBib.html#biBCCN85">[CCN+85, Chapter 7, Sections 3–5]</a>).

<code class="func">CambridgeMaps</code> is used for example by <code class="func">Display</code> (<a href="../../../doc/ref/chap71.html#X7B41F36478C47364">Reference: Display for a character table</a>) in the case that the <code class="code">powermap</code> option has the value <code class="code">"ATLAS"</code>.

Note that the value of the <code class="code">names</code> component may differ from the class names of the character table shown in the Atlas of Finite Groups; an example is the character table of the group J_1.

<div class="example"><pre>
gap> CambridgeMaps( CharacterTable( "A5" ) );
rec( names := [ "1A", "2A", "3A", "5A", "B*" ],
 power := [ "", "A", "A", "A", "A" ],
 prime := [ "", "A", "A", "A", "A" ] )
gap> CambridgeMaps( CharacterTable( "A5" ) mod 2 );
rec( names := [ "1A", "3A", "5A", "B*" ],
 power := [ "", "A", "A", "A" ], prime := [ "", "A", "A", "A" ] )
</pre></div>

<a id="X8209063C813240E5" name="X8209063C813240E5"></a>

<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">tblnames</var>[, <var class="Arg">p</var>] )</td><td class="tdright">( function )</td></tr></table></div>
Let <var class="Arg">tblnames</var> be a matrix of identifiers of ordinary character tables, which describe the bicyclic extensions of a simple group from the Atlas of Finite Groups. The class fusions between the character tables must be stored on the tables.

If the required information is available then <code class="func">StringOfCambridgeFormat</code> returns a string that encodes an approximation of the Cambridge format file for the simple group in question (whose identifier occurs in the upper left corner of <var class="Arg">tblnames</var>). Otherwise, that is, if some character table or class fusion is missing, <code class="keyw">fail</code> is returned.

If a prime integer <var class="Arg">p</var> is given as a second argument then the result describes <var class="Arg">p</var>-modular character tables, otherwise the ordinary character tables are described by the result.

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.

<div class="example"><pre>
gap> str:= StringOfCambridgeFormat( [ [ "A5", "A5.2" ],
> [ "2.A5", "2.A5.2" ] ] );;
gap> Print( str );
#23 ? A5
#7 4 4 4 4 4 4 4 4 4 4 4
#9 ; @ @ @ @ @ ; ; @ @ @
#1 | 60 4 3 5 5 | | 6 2 3
#2 p power A A A A | | A A AB
#3 p' part A A A A | | A A AB
#4 ind 1A 2A 3A 5A B* fus ind 2B 4A 6A
#5 + 1 1 1 1 1 : ++ 1 1 1
#5 + 3 -1 0 -b5 * . + 0 0 0
#5 + 3 -1 0 * -b5 . | | | |
#5 + 4 0 1 -1 -1 : ++ 2 0 -1
#5 + 5 1 -1 0 0 : ++ 1 -1 1
#6 ind 1 4 3 5 5 fus ind 2 8 6
#6 | 2 | 6 10 10 | | | 8 6
#5 - 2 0 -1 b5 * . - 0 0 0
#5 - 2 0 -1 * b5 . | | | |
#5 - 4 0 1 -1 -1 : oo 0 0 i3
#5 - 6 0 0 1 1 : oo 0 i2 0
#8
gap> str:= StringOfCambridgeFormat( [ [ "A5", "A5.2" ],
> [ "2.A5", "2.A5.2" ] ], 3 );;
gap> Print( str );
#23 A5 (Mod 3)
#7 4 4 4 4 4 4 4 4 4
#9 ; @ @ @ @ ; ; @ @
#1 | 60 4 5 5 | | 6 2
#2 p power A A A | | A A
#3 p' part A A A | | A A
#4 ind 1A 2A 5A B* fus ind 2B 4A
#5 + 1 1 1 1 : ++ 1 1
#5 + 3 -1 -b5 * . + 0 0
#5 + 3 -1 * -b5 . | | |
#5 + 4 0 -1 -1 : ++ 2 0
#6 ind 1 4 5 5 fus ind 2 8
#6 | 2 | 10 10 | | | 8
#5 - 2 0 b5 * . - 0 0
#5 - 2 0 * b5 . | | |
#5 - 6 0 1 1 : oo 0 i2
#8
gap> StringOfCambridgeFormat( [ [ "L10(11)" ] ], 0 );
fail
</pre></div>

The global option <code class="code">OmitDashedRows</code> can be used to control whether the two-line description of <q>dashed</q> row portions (concerning tables of, e. g., 2'.Sz(8)) are omitted (value true) or shown (value false). The default is to show information about dashed row portions in the case of ordinary tables, and to omit this information for Brauer tables.

<a id="X8676916F81F40A0F" name="X8676916F81F40A0F"></a>

<h4>6.5 Interface to the MAGMA System</h4>

This interface is intended to convert character tables given in MAGMA's (see [CP96]) display format into GAP character tables.

The function <code class="func">BosmaBase</code> (<a href="chap6.html#X79D160BD7ECA6D2F">6.5-1</a>) is used for the translation of irrational values; this function may be of interest independent of the conversion of character tables.

<a id="X79D160BD7ECA6D2F" name="X79D160BD7ECA6D2F"></a>

<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>
For a positive integer <var class="Arg">n</var> that is not congruent to 2 modulo 4, <code class="func">BosmaBase</code> returns the list of exponents i for which <code class="code">E(<var class="Arg">n</var>)^</code>i 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>.

As a set, this basis is defined as follows. Let P denote the set of prime divisors of <var class="Arg">n</var> and <var class="Arg">n</var> = ∏_{p ∈ P} n_p. Let e_l = <code class="code">E</code>(l) for any positive integer l, and { e_{m_1}^j }_{j ∈ J} ⊗ { e_{m_2}^k }_{k ∈ K} = { e_{m_1}^j ⋅ e_{m_2}^k }_{j ∈ J, k ∈ K} for any positive integers m_1, m_2. (This notation is the same as the one used in the description of <code class="func">ZumbroichBase</code> (<a href="../../../doc/ref/chap60.html#X7F52BEA0862E06F2">Reference: ZumbroichBase</a>).)

Then the basis is

B_n = ⨂_{p ∈ P} B_{n_p}

where

B_{n_p} = { e_{n_p}^k; 0 ≤ k ≤ φ(n_p)-1 };

here φ denotes Euler's function, see Phi (Reference: Phi).

B_n consists of roots of unity, it is an integral basis (that is, exactly the integral elements in ℚ_n have integral coefficients w.r.t. B_n, cf. <code class="func">IsIntegralCyclotomic</code> (<a href="../../../doc/ref/chap18.html#X869750DA81EA0E67">Reference: IsIntegralCyclotomic</a>)), and for any divisor m of <var class="Arg">n</var> that is not congruent to 2 modulo 4, B_m is a subset of B_n.

Note that the list l, say, that is returned by <code class="func">BosmaBase</code> is in general not a set. The ordering of the elements in l fits to the coefficient lists for irrational values used by MAGMA's display format.

<div class="example"><pre>
gap> b:= BosmaBase( 8 );
[ 0, 1, 2, 3 ]
gap> b:= Basis( CF(8), List( b, i -> E(8)^i ) );
Basis( CF(8), [ 1, E(8), E(4), E(8)^3 ] )
gap> Coefficients( b, Sqrt(2) );
[ 0, 1, 0, -1 ]
gap> Coefficients( b, Sqrt(-2) );
[ 0, 1, 0, 1 ]
gap> b:= BosmaBase( 15 );
[ 0, 5, 3, 8, 6, 11, 9, 14 ]
gap> b:= List( b, i -> E(15)^i );
[ 1, E(3), E(5), E(15)^8, E(5)^2, E(15)^11, E(5)^3, E(15)^14 ]
gap> Coefficients( Basis( CF(15), b ), EB(15) );
[ -1, -1, 0, 0, -1, -2, -1, -2 ]
gap> BosmaBase( 48 );
[ 0, 3, 6, 9, 12, 15, 18, 21, 16, 19, 22, 25, 28, 31, 34, 37 ]
</pre></div>

<a id="X7B1FAE1A7FD5D5A1" name="X7B1FAE1A7FD5D5A1"></a>

<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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GAPTableOfMagmaFile</code>( <var class="Arg">str</var>, <var class="Arg">identifier</var>[, <var class="Arg">"string"</var>] )</td><td class="tdright">( function )</td></tr></table></div>
In the first form, let <var class="Arg">file</var> be the name of a file that contains a character table in MAGMA's display format, and identifier be a string. GAPTableOfMagmaFile returns the corresponding GAP character table, with Identifier (Reference: Identifier for tables of marks) value identifier.

In the second form, <var class="Arg">str</var> must be a string that describes the contents of a file as described for the first form, and the third argument must be the string <code class="code">"string"</code>.

<div class="example"><pre>
gap> tmpdir:= DirectoryTemporary();;
gap> file:= Filename( tmpdir, "magmatable" );;
gap> str:= "\
> Character Table of Group G\n\
> --------------------------\n\
> \n\
> ---------------------------\n\
> Class | 1 2 3 4 5\n\
> Size | 1 15 20 12 12\n\
> Order | 1 2 3 5 5\n\
> ---------------------------\n\
> p = 2 1 1 3 5 4\n\
> p = 3 1 2 1 5 4\n\
> p = 5 1 2 3 1 1\n\
> ---------------------------\n\
> X.1 + 1 1 1 1 1\n\
> X.2 + 3 -1 0 Z1 Z1#2\n\
> X.3 + 3 -1 0 Z1#2 Z1\n\
> X.4 + 4 0 1 -1 -1\n\
> X.5 + 5 1 -1 0 0\n\
> \n\
> Explanation of Character Value Symbols\n\
> --------------------------------------\n\
> \n\
> # denotes algebraic conjugation, that is,\n\
> #k indicates replacing the root of unity w by w^k\n\
> \n\
> Z1 = (CyclotomicField(5: Sparse := true)) ! [\n\
> RationalField() | 1, 0, 1, 1 ]\n\
> ";;
gap> FileString( file, str );;
gap> tbl:= GAPTableOfMagmaFile( file, "MagmaA5" );;
gap> Display( tbl );
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
gap> tbl2:= GAPTableOfMagmaFile( str, "MagmaA5", "string" );;
gap> Irr( tbl ) = Irr( tbl2 );
true
gap> str:= "\
> Character Table of Group G\n\
> --------------------------\n\
> \n\
> ------------------------------\n\
> Class | 1 2 3 4 5 6\n\
> Size | 1 1 1 1 1 1\n\
> Order | 1 2 3 3 6 6\n\
> ------------------------------\n\
> p = 2 1 1 4 3 3 4\n\
> p = 3 1 2 1 1 2 2\n\
> ------------------------------\n\
> X.1 + 1 1 1 1 1 1\n\
> X.2 + 1 -1 1 1 -1 -1\n\
> X.3 0 1 1 J-1-J-1-J J\n\
> X.4 0 1 -1 J-1-J 1+J -J\n\
> X.5 0 1 1-1-J J J-1-J\n\
> X.6 0 1 -1-1-J J -J 1+J\n\
> \n\
> \n\
> Explanation of Character Value Symbols\n\
> --------------------------------------\n\
> \n\
> J = RootOfUnity(3)\n\
> ";;
gap> FileString( file, str );;
gap> tbl:= GAPTableOfMagmaFile( file, "MagmaC6" );;
gap> Display( tbl );
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>

The MAGMA output for the above two examples is obtained by the following commands.

<div class="example"><pre>
> G1 := Alt(5);
> CT1 := CharacterTable(G1);
> CT1;
> G2:= CyclicGroup(6);
> CT2:= CharacterTable(G2);
> CT2;
</pre></div>

<a id="X824D2B4A79A9E5AE" name="X824D2B4A79A9E5AE"></a>

<h5>6.5-3 CharacterTableComputedByMagma</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CharacterTableComputedByMagma</code>( <var class="Arg">G</var>, <var class="Arg">identifier</var> )</td><td class="tdright">( function )</td></tr></table></div>
For a permutation group <var class="Arg">G</var> and a string <var class="Arg">identifier</var>, <code class="func">CharacterTableComputedByMagma</code> calls the MAGMA system for computing the character table of <var class="Arg">G</var>, and converts the output into GAP format (see <code class="func">GAPTableOfMagmaFile</code> (<a href="chap6.html#X7B1FAE1A7FD5D5A1">6.5-2</a>)). The returned character table has the <code class="func">Identifier</code> (<a href="../../../doc/ref/chap70.html#X810E53597B5BB4F8">Reference: Identifier for tables of marks</a>) value <var class="Arg">identifier</var>.

If the MAGMA system is not available then <code class="keyw">fail</code> is returned. The availability of MAGMA is determined by calling MAGMA where the path for this call is given by the user preference <code class="code">MagmaPath</code> of the package CTblLib; if the value of this preference is empty or if MAGMA cannot be called via this path then MAGMA is regarded as not available.

If the attribute <code class="func">ConjugacyClasses</code> (<a href="../../../doc/ref/chap39.html#X871B570284BBA685">Reference: ConjugacyClasses attribute</a>) of <var class="Arg">G</var> is set before the call of <code class="func">CharacterTableComputedByMagma</code> then the columns of the returned character table fit to the conjugacy classes that are stored in <var class="Arg">G</var>.

<div class="example"><pre>
gap> if CTblLib.IsMagmaAvailable() then
> g:= MathieuGroup( 24 );
> ccl:= ConjugacyClasses( g );
> t:= CharacterTableComputedByMagma( g, "testM24" );
> if t = fail then
> Print( "#E Magma did not compute a character table.\n" );
> elif ( not HasConjugacyClasses( t ) ) or
> ( ConjugacyClasses( t ) <> ccl ) then
> Print( "#E The conjugacy classes do not fit.\n" );
> elif TransformingPermutationsCharacterTables( t,
> CharacterTable( "M24" ) ) = fail then
> Print( "#E Inconsistency of character tables?\n" );
> fi;
> fi;
</pre></div>

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