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<title> Decomposition Matrices in GAP</title>
<h1 align="center">Decomposition Matrices in GAP </h1>
<body bgcolor="FFFFFF">
<div class="p"><!----></div>
<h3 align="center"> T<font size="-2">HOMAS</font> B<font size="-2">REUER</font> <br />
<i>Lehrstuhl D für Mathematik</i> <br />
<i>RWTH, 52056 Aachen, Germany</i> </h3>
<div class="p"><!----></div>
<h3 align="center">June 13th, 1999 </h3>
<div class="p"><!----></div>
<div class="p"><!----></div>
This is a sample <font face="helvetica">GAP</font> session showing computations concerning
decomposition matrices.
For the description of the commands for character tables,
see the <font face="helvetica">GAP</font> Character Table Library [<a href="#CTblLib" name="CITECTblLib">Bre04</a>]
and the chapter "Character Tables" of the
<a href="link"><font face="helvetica">GAP</font> Reference Manual</a>.
<div class="p"><!----></div>
<div class="p"><!----></div>
<div class="p"><!----></div>
<pre>
gap> LoadPackage( "ctbllib" );
true
</pre>
<div class="p"><!----></div>
<h2><a name="tth_sEc1">
1</a> Basis Computations with Characters of M<sub>11</sub></h2>
<div class="p"><!----></div>
We start with the inspection of the Mathieu group M<sub>11</sub>.
Its ordinary character table and 2-modular Brauer table are fetched
from the character table library using the command <tt>CharacterTable</tt>
and the <tt>mod</tt> operator.
<div class="p"><!----></div>
<pre>
gap> ordtbl:= CharacterTable( "M11" );
CharacterTable( "M11" )
gap> p:= 2;
2
gap> modtbl:= ordtbl mod p;
BrauerTable( "M11", 2 )
</pre>
<div class="p"><!----></div>
The above commands assign the character tables to the variables <tt>ordtbl</tt>
and <tt>modtbl</tt>, respectively.
<div class="p"><!----></div>
The matrices of irreducible characters and additional information
such as centralizer orders (in factorized form) and power maps
are displayed using <tt>Display</tt>.
<div class="p"><!----></div>
<pre>
gap> Display( ordtbl );
M11
2 4 4 1 3 . 1 3 3 . .
3 2 1 2 . . 1 . . . .
5 1 . . . 1 . . . . .
11 1 . . . . . . . 1 1
1a 2a 3a 4a 5a 6a 8a 8b 11a 11b
2P 1a 1a 3a 2a 5a 3a 4a 4a 11b 11a
3P 1a 2a 1a 4a 5a 2a 8a 8b 11a 11b
5P 1a 2a 3a 4a 1a 6a 8b 8a 11a 11b
11P 1a 2a 3a 4a 5a 6a 8a 8b 1a 1a
X.1 1 1 1 1 1 1 1 1 1 1
X.2 10 2 1 2 . -1 . . -1 -1
X.3 10 -2 1 . . 1 A -A -1 -1
X.4 10 -2 1 . . 1 -A A -1 -1
X.5 11 3 2 -1 1 . -1 -1 . .
X.6 16 . -2 . 1 . . . B /B
X.7 16 . -2 . 1 . . . /B B
X.8 44 4 -1 . -1 1 . . . .
X.9 45 -3 . 1 . . -1 -1 1 1
X.10 55 -1 1 -1 . -1 1 1 . .
A = E(8)+E(8)^3
= ER(-2) = i2
B = E(11)+E(11)^3+E(11)^4+E(11)^5+E(11)^9
= (-1+ER(-11))/2 = b11
gap> Display( modtbl );
M11mod2
2 4 1 . . .
3 2 2 . . .
5 1 . 1 . .
11 1 . . 1 1
1a 3a 5a 11a 11b
2P 1a 3a 5a 11b 11a
3P 1a 1a 5a 11a 11b
5P 1a 3a 1a 11a 11b
11P 1a 3a 5a 1a 1a
X.1 1 1 1 1 1
X.2 10 1 . -1 -1
X.3 16 -2 1 A /A
X.4 16 -2 1 /A A
X.5 44 -1 -1 . .
A = E(11)+E(11)^3+E(11)^4+E(11)^5+E(11)^9
= (-1+ER(-11))/2 = b11
</pre>
<div class="p"><!----></div>
The restrictions of ordinary irreducible characters to
conjugacy classes of element order prime to p decompose into the
irreducible p-modular Brauer characters,
with nonnegative integer coefficients.
The matrix of coefficients, the so-called decomposition matrix,
w.r.t. the given ordering of characters is computed by <tt>DecompositionMatrix</tt>.
Again <tt>Display</tt> can be used to produce a formatted output.
<div class="p"><!----></div>
<pre>
gap> mat:= DecompositionMatrix( modtbl );
[ [ 1, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ],
[ 1, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ], [ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ],
[ 1, 0, 0, 0, 1 ], [ 1, 1, 0, 0, 1 ] ]
gap> Display( mat );
[ [ 1, 0, 0, 0, 0 ],
[ 0, 1, 0, 0, 0 ],
[ 0, 1, 0, 0, 0 ],
[ 0, 1, 0, 0, 0 ],
[ 1, 1, 0, 0, 0 ],
[ 0, 0, 1, 0, 0 ],
[ 0, 0, 0, 1, 0 ],
[ 0, 0, 0, 0, 1 ],
[ 1, 0, 0, 0, 1 ],
[ 1, 1, 0, 0, 1 ] ]
</pre>
<div class="p"><!----></div>
Often one is more interested in the decomposition matrices of a specific
p-block.
The distribution of ordinary irreducible characters to blocks is computed
by <tt>PrimeBlocks</tt>.
<div class="p"><!----></div>
<pre>
gap> blocks:= PrimeBlocks( ordtbl, p );;
gap> blocks.block;
[ 1, 1, 1, 1, 1, 2, 3, 1, 1, 1 ]
gap> blocks.defect;
[ 4, 0, 0 ]
</pre>
<div class="p"><!----></div>
The output means that the first five and the last three irreducibles
of M<sub>11</sub> together lie in the principal block, which has defect 4,
and the remaining two irreducibles are defect zero characters.
<div class="p"><!----></div>
This information is computed from the ordinary table only,
so it is available also if the Brauer table is not known.
But if we have access to the brauer table then information concerning
the dirstribution of ordinary and Brauer characters to blocks is
computed with <tt>BlocksInfo</tt>, applied to the Brauer table.
<div class="p"><!----></div>
<pre>
gap> blocksinfo:= BlocksInfo( modtbl );
[ rec( defect := 4, ordchars := [ 1, 2, 3, 4, 5, 8, 9, 10 ],
modchars := [ 1, 2, 5 ],
decinv := [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
basicset := [ 1, 2, 8 ] ),
rec( defect := 0, ordchars := [ 6 ], modchars := [ 3 ], decinv := [ [ 1 ] ],
basicset := [ 6 ] ),
rec( defect := 0, ordchars := [ 7 ], modchars := [ 4 ], decinv := [ [ 1 ] ],
basicset := [ 7 ] ) ]
</pre>
<div class="p"><!----></div>
If we are interested in the decomposition matrix of the i-th block
then i can be entered as second parameter of the command
<tt>DecompositionMatrix</tt>.
So the following command computes and displays the decomposition matrix
of the principal block.
<div class="p"><!----></div>
<pre>
gap> Display( DecompositionMatrix( modtbl, 1 ) );
[ [ 1, 0, 0 ],
[ 0, 1, 0 ],
[ 0, 1, 0 ],
[ 0, 1, 0 ],
[ 1, 1, 0 ],
[ 0, 0, 1 ],
[ 1, 0, 1 ],
[ 1, 1, 1 ] ]
</pre>
<div class="p"><!----></div>
The return value of <tt>BlocksInfo</tt> is a list, the i-th entry being a record
that comprises the data about the i-th block.
The record for the principal block can be accessed as follows.
<div class="p"><!----></div>
<pre>
gap> principalinfo:= blocksinfo[1];
rec( defect := 4, ordchars := [ 1, 2, 3, 4, 5, 8, 9, 10 ],
modchars := [ 1, 2, 5 ], decinv := [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ]
, basicset := [ 1, 2, 8 ],
decmat := [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 1, 0 ], [ 0, 1, 0 ],
[ 1, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 1 ], [ 1, 1, 1 ] ] )
</pre>
<div class="p"><!----></div>
We see that additionally to the <tt>blocksinfo</tt> value we had got above,
now the decomposition matrix of the block is stored in the record.
This is because <tt>DecompositionMatrix</tt> has stored the matrix once
it had been computed.
<div class="p"><!----></div>
Components of the record can be accessed as follows.
<div class="p"><!----></div>
<pre>
gap> ordpos:= principalinfo.ordchars;
[ 1, 2, 3, 4, 5, 8, 9, 10 ]
</pre>
<div class="p"><!----></div>
This list is the list of positions of the characters in the principal
block, w.r.t. the ordering in <tt>ordtbl</tt>.
The ordinary irreducibles in the principal block can then be extracted
using the <tt>Irr</tt> command as follows.
<div class="p"><!----></div>
<pre>
gap> ordchars:= Irr( ordtbl ){ ordpos };
[ Character( CharacterTable( "M11" ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ),
Character( CharacterTable( "M11" ), [ 10, 2, 1, 2, 0, -1, 0, 0, -1, -1 ] ),
Character( CharacterTable( "M11" ), [ 10, -2, 1, 0, 0, 1, E(8)+E(8)^3,
-E(8)-E(8)^3, -1, -1 ] ), Character( CharacterTable( "M11" ),
[ 10, -2, 1, 0, 0, 1, -E(8)-E(8)^3, E(8)+E(8)^3, -1, -1 ] ),
Character( CharacterTable( "M11" ), [ 11, 3, 2, -1, 1, 0, -1, -1, 0, 0 ] ),
Character( CharacterTable( "M11" ), [ 44, 4, -1, 0, -1, 1, 0, 0, 0, 0 ] ),
Character( CharacterTable( "M11" ), [ 45, -3, 0, 1, 0, 0, -1, -1, 1, 1 ] ),
Character( CharacterTable( "M11" ), [ 55, -1, 1, -1, 0, -1, 1, 1, 0, 0 ] ) ]
</pre>
<div class="p"><!----></div>
We restrict these characters to the p-regular classes,
and use <tt>Decomposition</tt> to compute the decomposition of these characters
into the Brauer characters of the principal block.
The result is again the decomposition matrix,
and this is exactly what <tt>DecompositionMatrix</tt> does.
<div class="p"><!----></div>
<pre>
gap> rest:= RestrictedClassFunctions( ordchars, modtbl );
[ Character( BrauerTable( "M11", 2 ), [ 1, 1, 1, 1, 1 ] ),
Character( BrauerTable( "M11", 2 ), [ 10, 1, 0, -1, -1 ] ),
Character( BrauerTable( "M11", 2 ), [ 10, 1, 0, -1, -1 ] ),
Character( BrauerTable( "M11", 2 ), [ 10, 1, 0, -1, -1 ] ),
Character( BrauerTable( "M11", 2 ), [ 11, 2, 1, 0, 0 ] ),
Character( BrauerTable( "M11", 2 ), [ 44, -1, -1, 0, 0 ] ),
Character( BrauerTable( "M11", 2 ), [ 45, 0, 0, 1, 1 ] ),
Character( BrauerTable( "M11", 2 ), [ 55, 1, 0, 0, 0 ] ) ]
gap> modchars:= Irr( modtbl ){ principalinfo.modchars };
[ Character( BrauerTable( "M11", 2 ), [ 1, 1, 1, 1, 1 ] ),
Character( BrauerTable( "M11", 2 ), [ 10, 1, 0, -1, -1 ] ),
Character( BrauerTable( "M11", 2 ), [ 44, -1, -1, 0, 0 ] ) ]
gap> dec:= Decomposition( modchars, rest, "nonnegative" );
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 1, 0 ], [ 0, 1, 0 ], [ 1, 1, 0 ],
[ 0, 0, 1 ], [ 1, 0, 1 ], [ 1, 1, 1 ] ]
</pre>
<div class="p"><!----></div>
For printing decomposition matrices, <font face="helvetica">GAP</font> provides the function
<tt>LaTeXStringDecompositionMatrix</tt> that produces <span class="roman">L</span><sup><span class="roman">A</span></sup><span class="roman">T</span><sub><span class="roman">E</span></sub><span class="roman">X</span> code.
(If the second argument is omitted then <span class="roman">L</span><sup><span class="roman">A</span></sup><span class="roman">T</span><sub><span class="roman">E</span></sub><span class="roman">X</span> code for printing
the full decomposition matrix of the group is produced.)
<div class="p"><!----></div>
<pre>
gap> Print( LaTeXStringDecompositionMatrix( modtbl, 1 ) );
\[
\begin{array}{r|rrr} \hline
& {\tt Y}_{1}
& {\tt Y}_{2}
& {\tt Y}_{5}
\rule[-7pt]{0pt}{20pt} \\ \hline
{\tt X}_{1} & 1 & . & . \rule[0pt]{0pt}{13pt} \\
{\tt X}_{2} & . & 1 & . \\
{\tt X}_{3} & . & 1 & . \\
{\tt X}_{4} & . & 1 & . \\
{\tt X}_{5} & 1 & 1 & . \\
{\tt X}_{8} & . & . & 1 \\
{\tt X}_{9} & 1 & . & 1 \\
{\tt X}_{10} & 1 & 1 & 1 \rule[-7pt]{0pt}{5pt} \\
\hline
\end{array}
\]
</pre>
<div class="p"><!----></div>
In a <span class="roman">L</span><sup><span class="roman">A</span></sup><span class="roman">T</span><sub><span class="roman">E</span></sub><span class="roman">X</span> document, this looks as follows.
<div class="p"><!----></div>
<br clear="all" /><table border="0" width="100%"><tr><td>
<table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center">
</td><td nowrap="nowrap" align="center">
<table>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
</td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
<tt>Y</tt><sub>1</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
<tt>Y</tt><sub>2</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
<tt>Y</tt><sub>5</sub> </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
<tt>X</tt><sub>1</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
<tt>X</tt><sub>2</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
<tt>X</tt><sub>3</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
<tt>X</tt><sub>4</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
<tt>X</tt><sub>5</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
<tt>X</tt><sub>8</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
<tt>X</tt><sub>9</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
<tt>X</tt><sub>10</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td></tr></table>
</td><td nowrap="nowrap" align="center">
</td></tr></table>
</td></tr></table>
<div class="p"><!----></div>
<h2><a name="tth_sEc2">
2</a> Blocks of Symmetric Groups</h2>
<div class="p"><!----></div>
The conjugacy classes and the irreducibles characters of the symmetric
group of degree n, say,
correspond to the partitions of n in a natural way,
and it is reasonable to use these partitions as labels for the (ordinary)
irreducibles in decomposition matrices.
<div class="p"><!----></div>
<div class="p"><!----></div>
First we need a little utility for converting a list denoting a partition
into a short string.
For example, <tt>[ 2, 1, 1, 1, 1 ]</tt> shall be written as 2 1<sup>4</sup>.
<div class="p"><!----></div>
<pre>
gap> StringOfPartition:= function( part )
> local pair;
>
> part:= List( Reversed( Collected( part ) ),
> pair -> List( pair, String ) );
> for pair in part do
> if pair[2] = "1" then
> Unbind( pair[2] );
> else
> pair[2]:= Concatenation( "{", pair[2], "}" );
> fi;
> od;
> return JoinStringsWithSeparator( List( part,
> p -> JoinStringsWithSeparator( p, "^" ) ), " \\ " );
> end;;
gap> StringOfPartition( [ 2, 1, 1, 1, 1 ] );
"2 \\ 1^{4}"
</pre>
<div class="p"><!----></div>
Now we put also the construction of the row labels and of the final string
into a little function.
Here we use that the partitions corresponding to the ordinary irreducibles
of a character table of a symmetric group are stored in the attribute
<tt>CharacterParameters</tt>, provided that the table was constructed with
<tt>CharacterTable( "Symmetric", n )</tt> and not from the symmetric group;
for small values of n, for example n = 7, also <tt>CharacterTable( "S7" )</tt>
can be used.
Each entry in the <tt>CharacterParameters</tt> list for the table of a symmetric
group is a pair, the desired partition is the second entry.
<div class="p"><!----></div>
<pre>
gap> MyLaTeXMatrix:= function( symt, blocknr )
> local alllabels, rowlabels;
>
> alllabels:= List( CharacterParameters( OrdinaryCharacterTable( symt ) ),
> x -> StringOfPartition( x[2] ) );
> rowlabels:= alllabels{ BlocksInfo( symt )[ blocknr ].ordchars };
>
> return LaTeXStringDecompositionMatrix( symt, blocknr,
> rec( phi:= "\\varphi", rowlabels:= rowlabels ) );
> end;;
</pre>
<div class="p"><!----></div>
Now we get for example the following output for the principal block of the
5-modular table of S<sub>6</sub>.
<div class="p"><!----></div>
<pre>
gap> t:= CharacterTable( "S6" ) mod 5;
BrauerTable( "A6.2_1", 5 )
gap> b:= 1;
1
gap> Print( MyLaTeXMatrix( t, b ) );
\[
\begin{array}{r|rrrr} \hline
& \varphi_{1}
& \varphi_{2}
& \varphi_{7}
& \varphi_{8}
\rule[-7pt]{0pt}{20pt} \\ \hline
6 & 1 & . & . & . \rule[0pt]{0pt}{13pt} \\
1^{6} & . & 1 & . & . \\
3 \ 2 \ 1 & . & . & 1 & 1 \\
4 \ 2 & 1 & . & 1 & . \\
2^{2} \ 1^{2} & . & 1 & . & 1 \rule[-7pt]{0pt}{5pt} \\
\hline
\end{array}
\]
</pre>
<div class="p"><!----></div>
In the <span class="roman">L</span><sup><span class="roman">A</span></sup><span class="roman">T</span><sub><span class="roman">E</span></sub><span class="roman">X</span> document, this looks as follows.
<div class="p"><!----></div>
<br clear="all" /><table border="0" width="100%"><tr><td>
<table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center">
</td><td nowrap="nowrap" align="center">
<table>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
</td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
ϕ<sub>1</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
ϕ<sub>2</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
ϕ<sub>7</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
ϕ<sub>8</sub> </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
6 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1<sup>6</sup> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
3 2 1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
4 2 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
2<sup>2</sup> 1<sup>2</sup> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td></tr></table>
</td><td nowrap="nowrap" align="center">
</td></tr></table>
</td></tr></table>
<div class="p"><!----></div>
For breaking the array into several row and column portions,
we could put the optional components <tt>nrows</tt> and <tt>ncols</tt> into the record
that is given as the second argument of <tt>LaTeXStringDecompositionMatrix</tt>.
<div class="p"><!----></div>
<h2><a name="tth_sEc3">
3</a> Computations with A<font size="-2">TLAS</font> Tables in <font face="helvetica">GAP</font></h2>
<div class="p"><!----></div>
When dealing with ordinary character tables contained in the A<font size="-2">TLAS</font> of
Finite Groups and Brauer character tables contained in the A<font size="-2">TLAS</font> of
Brauer Characters,
it is convenient to refer to characters and conjugacy classes via the
labels chosen in these books.
<div class="p"><!----></div>
For a simple group G, the ordering of characters and classes in the
A<font size="-2">TLAS</font> table and in the <font face="helvetica">GAP</font> library table coincide.
This means that the ordinary irreducible character with A<font size="-2">TLAS</font> label
χ<sub>i</sub> is the i-th entry in the list of irreducibles returned by
the function <tt>Irr</tt> when called with the ordinary <font face="helvetica">GAP</font> library table.
Analogously, the j-th entry in the <tt>Irr</tt> list of the p-modular
<font face="helvetica">GAP</font> library table has A<font size="-2">TLAS</font> label ϕ<sub>j</sub>.
<div class="p"><!----></div>
The function <tt>AtlasLabelsOfIrreducibles</tt> returns the list of A<font size="-2">TLAS</font>
labels of a <font face="helvetica">GAP</font> library table.
<div class="p"><!----></div>
<pre>
gap> ordtbl:= CharacterTable( "A6" );
CharacterTable( "A6" )
gap> ordlabels:= AtlasLabelsOfIrreducibles( ordtbl );
[ "\\chi_{1}", "\\chi_{2}", "\\chi_{3}", "\\chi_{4}", "\\chi_{5}",
"\\chi_{6}", "\\chi_{7}" ]
gap> modtbl:= ordtbl mod 5;
BrauerTable( "A6", 5 )
gap> modlabels:= AtlasLabelsOfIrreducibles( modtbl );
[ "\\varphi_{1}", "\\varphi_{2}", "\\varphi_{3}", "\\varphi_{4}",
"\\varphi_{5}" ]
</pre>
<div class="p"><!----></div>
These labels can be used to replace the default labels used by
<tt>LaTeXStringDecompositionMatrix</tt>, via an optional record entered as
third argument.
<div class="p"><!----></div>
<pre>
gap> rowlabels:= ordlabels{ BlocksInfo( modtbl )[1].ordchars };
[ "\\chi_{1}", "\\chi_{4}", "\\chi_{5}", "\\chi_{6}" ]
gap> collabels:= modlabels{ BlocksInfo( modtbl )[1].modchars };
[ "\\varphi_{1}", "\\varphi_{4}" ]
gap> options:= rec( rowlabels:= rowlabels, collabels:= collabels );;
gap> Print( LaTeXStringDecompositionMatrix( modtbl, 1, options ) );
\[
\begin{array}{r|rr} \hline
& \varphi_{1}
& \varphi_{4}
\rule[-7pt]{0pt}{20pt} \\ \hline
\chi_{1} & 1 & . \rule[0pt]{0pt}{13pt} \\
\chi_{4} & . & 1 \\
\chi_{5} & . & 1 \\
\chi_{6} & 1 & 1 \rule[-7pt]{0pt}{5pt} \\
\hline
\end{array}
\]
</pre>
<div class="p"><!----></div>
In the <span class="roman">L</span><sup><span class="roman">A</span></sup><span class="roman">T</span><sub><span class="roman">E</span></sub><span class="roman">X</span> document, this looks as follows.
<div class="p"><!----></div>
<br clear="all" /><table border="0" width="100%"><tr><td>
<table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center">
</td><td nowrap="nowrap" align="center">
<table>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
</td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
ϕ<sub>1</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
ϕ<sub>4</sub> </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
χ<sub>1</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
χ<sub>4</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
χ<sub>5</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
χ<sub>6</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td></tr></table>
</td><td nowrap="nowrap" align="center">
</td></tr></table>
</td></tr></table>
<div class="p"><!----></div>
The output can be influenced by several other components of the
options record, for example bigger matrices can be broken into
several portions of rows and columns using components <tt>nrows</tt> and
<tt>ncols</tt>.
<div class="p"><!----></div>
For central extensions of simple groups by a group of order at least 3,
the ordering of characters in the A<font size="-2">TLAS</font> table and the <font face="helvetica">GAP</font> table
differ in the sense that the A<font size="-2">TLAS</font> does not list all irreducibles.
In the list of labels of the <font face="helvetica">GAP</font> table, the characters not printed in
the A<font size="-2">TLAS</font> are denoted as algebraic conjugates of characters printed in
the A<font size="-2">TLAS</font>.
<div class="p"><!----></div>
<pre>
gap> AtlasLabelsOfIrreducibles( CharacterTable( "3.A6" ) );
[ "\\chi_{1}", "\\chi_{2}", "\\chi_{3}", "\\chi_{4}", "\\chi_{5}",
"\\chi_{6}", "\\chi_{7}", "\\chi_{14}", "\\chi_{14}^{\\ast 11}",
"\\chi_{15}", "\\chi_{15}^{\\ast 11}", "\\chi_{16}", "\\chi_{16}^{\\ast 2}",
"\\chi_{17}", "\\chi_{17}^{\\ast 2}", "\\chi_{18}", "\\chi_{18}^{\\ast 2}" ]
gap> AtlasLabelsOfIrreducibles( CharacterTable( "3.A6" ) mod 5 );
[ "\\varphi_{1}", "\\varphi_{2}", "\\varphi_{3}", "\\varphi_{4}",
"\\varphi_{5}", "\\varphi_{10}", "\\varphi_{10}^{\\ast 2}", "\\varphi_{11}",
"\\varphi_{11}^{\\ast 2}", "\\varphi_{12}", "\\varphi_{12}^{\\ast 2}" ]
</pre>
<div class="p"><!----></div>
Note that due to the numbering of characters in the A<font size="-2">TLAS</font>,
certain "offsets" in the labels of a table may occur.
For example, the first faithful ordinary character of 3.A<sub>6</sub> has
number 14 in the A<font size="-2">TLAS</font> but is the 8-th in the <font face="helvetica">GAP</font> table,
and the 9-th character in the <font face="helvetica">GAP</font> table is the complex conjugate
of this character, which is not printed in the A<font size="-2">TLAS</font>.
<div class="p"><!----></div>
For cyclic upward extensions of perfect groups,
the labels of irreducibles are formed relative to those of the
derived subgroup.
Namely, extensions of a character with label χ<sub>i</sub> have labels of the
form χ<sub>i,j</sub>, with nonnegative integers j,
and an irreducible character whose restriction to the derived subgroup
is the sum of pairwise different characters χ<sub>i<sub>1</sub></sub>, χ<sub>i<sub>2</sub></sub>,
..., χ<sub>i<sub>k</sub></sub> has label χ<sub>i<sub>1</sub> + i<sub>2</sub> + …+ i<sub>k</sub></sub>;
for the latter kind of labels, a short form χ<sub>i<sub>1</sub>+</sub> are available.
<div class="p"><!----></div>
<pre>
gap> AtlasLabelsOfIrreducibles( CharacterTable( "3.A6.2_1" ) );
[ "\\chi_{1,0}", "\\chi_{1,1}", "\\chi_{2,0}", "\\chi_{2,1}", "\\chi_{3,0}",
"\\chi_{3,1}", "\\chi_{4+5}", "\\chi_{6,0}", "\\chi_{6,1}", "\\chi_{7,0}",
"\\chi_{7,1}", "\\chi_{14+15\\ast 11}", "\\chi_{14\\ast 11+15}",
"\\chi_{16+16\\ast 2}", "\\chi_{17+17\\ast 2}", "\\chi_{18+18\\ast 2}" ]
gap> AtlasLabelsOfIrreducibles( CharacterTable( "3.A6.2_1" ), "short" );
[ "\\chi_{1,0}", "\\chi_{1,1}", "\\chi_{2,0}", "\\chi_{2,1}", "\\chi_{3,0}",
"\\chi_{3,1}", "\\chi_{4+}", "\\chi_{6,0}", "\\chi_{6,1}", "\\chi_{7,0}",
"\\chi_{7,1}", "\\chi_{14+}", "\\chi_{15+}", "\\chi_{16+}", "\\chi_{17+}",
"\\chi_{18+}" ]
</pre>
<div class="p"><!----></div>
Used for printing decomposition matrices, these labels occur as follows.
<div class="p"><!----></div>
<pre>
gap> ordtbl:= CharacterTable( "3.A6.2_1" );;
gap> ordlabels:= AtlasLabelsOfIrreducibles( ordtbl, "short" );;
gap> modtbl:= ordtbl mod 3;;
gap> modlabels:= AtlasLabelsOfIrreducibles( modtbl, "short" );;
gap> rowlabels:= ordlabels{ BlocksInfo( modtbl )[1].ordchars };;
gap> collabels:= modlabels{ BlocksInfo( modtbl )[1].modchars };;
gap> options:= rec( rowlabels:= rowlabels, collabels:= collabels );;
gap> Print( LaTeXStringDecompositionMatrix( modtbl, 1, options ) );
\[
\begin{array}{r|rrrrr} \hline
& \varphi_{1,0}
& \varphi_{1,1}
& \varphi_{2+}
& \varphi_{4,0}
& \varphi_{4,1}
\rule[-7pt]{0pt}{20pt} \\ \hline
\chi_{1,0} & 1 & . & . & . & . \rule[0pt]{0pt}{13pt} \\
\chi_{1,1} & . & 1 & . & . & . \\
\chi_{2,0} & 1 & . & . & 1 & . \\
\chi_{2,1} & . & 1 & . & . & 1 \\
\chi_{3,0} & 1 & . & . & . & 1 \\
\chi_{3,1} & . & 1 & . & 1 & . \\
\chi_{4+} & 1 & 1 & 1 & 1 & 1 \\
\chi_{7,0} & . & . & 1 & 1 & . \\
\chi_{7,1} & . & . & 1 & . & 1 \\
\chi_{14+} & . & . & 1 & . & . \\
\chi_{15+} & . & . & 1 & . & . \\
\chi_{16+} & 2 & 2 & . & 1 & 1 \\
\chi_{18+} & 1 & 1 & 2 & 2 & 2 \rule[-7pt]{0pt}{5pt} \\
\hline
\end{array}
\]
</pre>
<div class="p"><!----></div>
In the <span class="roman">L</span><sup><span class="roman">A</span></sup><span class="roman">T</span><sub><span class="roman">E</span></sub><span class="roman">X</span> document, this looks as follows.
<div class="p"><!----></div>
<br clear="all" /><table border="0" width="100%"><tr><td>
<table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center">
</td><td nowrap="nowrap" align="center">
<table>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
</td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
ϕ<sub>1,0</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
ϕ<sub>1,1</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
ϕ<sub>2+</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
ϕ<sub>4,0</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
ϕ<sub>4,1</sub> </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
χ<sub>1,0</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
χ<sub>1,1</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
χ<sub>2,0</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
χ<sub>2,1</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
χ<sub>3,0</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
χ<sub>3,1</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
χ<sub>4+</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
χ<sub>7,0</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
χ<sub>7,1</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
χ<sub>14+</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
χ<sub>15+</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
χ<sub>16+</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
2 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
2 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
. </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td></tr>
<tr><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
χ<sub>18+</sub> </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
1 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
2 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
2 </td></tr></table></td><td align="right"><table border="0" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center">
2 </td></tr></table></td></tr></table>
</td><td nowrap="nowrap" align="center">
</td></tr></table>
</td></tr></table>
<div class="p"><!----></div>
For further questions about <font face="helvetica">GAP</font>, consult the <font face="helvetica">GAP</font> Reference Manual.
<div class="p"><!----></div>
<h2>References</h2>
<dl compact="compact">
<dt><a href="#CITECTblLib" name="CTblLib">[Bre04]</a></dt><dd>
Thomas Breuer, <em>Manual for the <font face="helvetica">GAP</font> Character Table Library, Version
1.1</em>, Lehrstuhl D für Mathematik, Rheinisch
Westfälische Technische Hochschule, Aachen, Germany,
2004.
<div class="p"><!----></div>
</dd>
<dt><a href="#CITEGAP4" name="GAP4">[GAP04]</a></dt><dd>
The GAP Group, <em>GAP - Groups, Algorithms, and Programming, Version
4.4</em>, 2004, <a href="http://www.gap-system.org"><tt>http://www.gap-system.org</tt></a>.</dd>
</dl>
<div class="p"><!----></div>
<div class="p"><!----></div>
<br /><br /><hr /><small>File translated from
T<sub><font size="-1">E</font></sub>X
by <a href="http://hutchinson.belmont.ma.us/tth/">
T<sub><font size="-1">T</font></sub>H</a>,
version 3.55.<br />On 31 Mar 2004, 10:53.</small>
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