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<p><a id="X8145828B844789C2" name="X8145828B844789C2"></a></p>
<div class="ChapSects"><a href="chap2.html#X8145828B844789C2">2 <span class="Heading">Irreducible Representations</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X812E7C8E7FA7BE4E">2.1 <span class="Heading">Constructing Representations</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7A3ACAC179C15393">2.1-1 IrreducibleAffordingRepresentation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7CF2E8907C5C6D0D">2.1-2 IsAffordingRepresentation</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X7EE113967DB4AB68">2.2 <span class="Heading">Induction</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X832316958253A958">2.2-1 InducedSubgroupRepresentation</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X7B3BE908867CE4F9">2.3 <span class="Heading">Extension</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7950DD2D7B80E583">2.3-1 ExtendedRepresentation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X83536FA985B4E255">2.3-2 ExtendedRepresentationNormal</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X80C4BBFE7CDE7AA9">2.4 <span class="Heading">Character Subgroups</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X81E72F66831F159E">2.4-1 CharacterSubgroupRepresentation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X79510CED811937D9">2.4-2 IsCharacterSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8408A559823FED05">2.4-3 AllCharacterPSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X86388596828FDAFE">2.4-4 AllCharacterStandardSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7A7B4ECC8722140C">2.4-5 AllCharacterSubgroups</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X8459AEDB7B73584F">2.5 <span class="Heading">Equivalent Representation</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X87C25B877C4952BA">2.5-1 EquivalentRepresentation</a></span>
</div>
</div>
<h3>2 <span class="Heading">Irreducible Representations</span></h3>
<p>Let G be a finite group and chi be an ordinary irreducible character of G. In this chapter we introduce some functions to construct a complex representation R of G affording chi. We proceed recursively, reducing the problem to smaller subgroups of G or characters of smaller degree until we obtain a problem which we can deal with directly. Inputs of most of the functions are a given group G, and an irreducible character chi. The output is a mapping (representation) which assigns to each generator x of G a matrix R(x). We can use these functions for all groups and all irreducible characters chi of degree less than 100 although in principle the same methods can be extended to characters of larger degree. The main methods in these functions which are used to construct representations of finite groups are Induction, Extension, Tensor Product and Dixon's method (for constructing representations of simple groups and their covers) <a href="chapBib.html#biBDab-05">[Dab-05]</a>, and Projective Representation method <a href="chapBib.html#biBDD-08">[DD-08]</a>.</p>
<p><a id="X812E7C8E7FA7BE4E" name="X812E7C8E7FA7BE4E"></a></p>
<h4>2.1 <span class="Heading">Constructing Representations</span></h4>
<p>This section introduces the main function to compute a representation of a finite group G affording an irreducible character chi of G.</p>
<p><a id="X7A3ACAC179C15393" name="X7A3ACAC179C15393"></a></p>
<h5>2.1-1 IrreducibleAffordingRepresentation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> IrreducibleAffordingRepresentation</code>( <var class="Arg">chi</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>called with an irreducible character <var class="Arg">chi</var> of a group G, this function returns a mapping (representation) which maps each generator of G to a d*d matrix, where d is the degree of <var class="Arg">chi</var>. The group generated by these matrices (the image of the map) is a matrix group which is isomorphic to G modulo the kernel of the map. If G is a solvable group then there is no restriction on the degree of <var class="Arg">chi</var>. In the case that G is not solvable and the character <var class="Arg">chi</var> has degree bigger than 100 the output maybe is not correct. In this case sometimes the output mapping does not afford the given character or it does not return any mapping.</p>
<table class="example">
<tr><td><pre>
gap> s := PerfectGroup( 129024, 2 );;
gap> G := Image(IsomorphismPermGroup( s ));;
gap> chi := Irr( G )[36];;
gap> chi[1];
64
gap> IrreducibleAffordingRepresentation( chi );;
#I Warning: EpimorphismSchurCover via Holt's algorithm is under construction
gap> time;
92657
</pre></td></tr></table>
<p><a id="X7CF2E8907C5C6D0D" name="X7CF2E8907C5C6D0D"></a></p>
<h5>2.1-2 IsAffordingRepresentation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> IsAffordingRepresentation</code>( <var class="Arg">chi, rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>If <var class="Arg">chi</var> and <var class="Arg">rep</var> are a character and a representation of a group G, respectively, then <code class="code">IsAffordingRepresentation</code> returns <code class="code">true</code> if the trace of <var class="Arg">rep(x)</var> equals <var class="Arg">chi(x)</var> for all elements x in G.</p>
<table class="example">
<tr><td><pre>
gap> G := GL(2,7);:
gap> chi := Irr(G)[ 29 ];;
gap> rep := IrreducibleAffordingRepresentation( chi );
CompositionMapping( [(8,15,22,29,36,43)(9,16,23,30,37,44)
(10,17,24,31,38,45)(11,18,25,32,39,46)(12,19,26,33,40,47)
(13,20,27,34,41,48)(14,21,28,35,42,49), (2,29,12)(3,36,20)
(4,43,28)(5,8,30)(6,15,38)(7,22,46)(9,44,14)(10,16,17)
(11,37,27)(13,23,39)(18,24,25)(19,45,35)(21,31,47)
(26,32,33)(34,40,41)(42,48,49) ] ->
[ [ [ 0, 0, 0, -1, 0, 0, 0 ],
[ 1, 0, -1, -1, 1, 0, -1 ]
[ 2, -1, -2, -2, 1, 2, -1 ],
[ 0, 0, -1, 0, 0, 0, 0 ],
[ 1, 0, -2, 0, 0, 1, -1 ],
[ 1, 0, -2, -1, 1, 1, -1 ],
[ -2, 1, 1, 1, -1, -1, 0 ] ],
[ [ 1, -1, -1, -1, 0, 2, -1 ],
[ 0, 0, 1, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 1, 0 ],
[ 0, 1, -1, 0, 0, 0, -1 ],
[ 0, 1, 0, 1, 0, -1, 0 ],
[ 0, 1, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, -1, 0, 0 ] ] ], (action isomorphism) )
gap> IsAffordingRepresentation( chi, rep );
true
</pre></td></tr></table>
<p>We can obtain the size of the image of this representation by <code class="code">Size(Image(rep))</code> and compute the value for an arbitrary element x in G by <code class="code">x</code>^<code class="code">rep</code>.</p>
<p><a id="X7EE113967DB4AB68" name="X7EE113967DB4AB68"></a></p>
<h4>2.2 <span class="Heading">Induction</span></h4>
<p><a id="X832316958253A958" name="X832316958253A958"></a></p>
<h5>2.2-1 InducedSubgroupRepresentation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> InducedSubgroupRepresentation</code>( <var class="Arg">G, rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>computes a representation of <var class="Arg">G</var> induced from the representation <var class="Arg">rep</var> of a subgroup H of <var class="Arg">G</var>. If <var class="Arg">rep</var> has degree d then the degree of the output representation is d*|G:H|.</p>
<table class="example">
<tr><td><pre>
gap> G := SymmetricGroup( 6 );;
gap> H := AlternatingGroup( 6 );;
gap> chi := Irr( H )[ 2 ];;
gap> rep := IrreducibleAffordingRepresentation( chi );;
gap> InducedSubgroupRepresentation( G, rep );
[ (1,2,3,4,5,6), (1,2) ] ->
[ [ [ 0, 0, 0, 0, 0, 1, 1, -1, -1, -1 ],
[ 0, 0, 0, 0, 0, 1, 0, -1, 0, -1 ],
[ 0, 0, 0, 0, 0, 1, 0, 0, -1, -1 ],
[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 1, -1, 0, -1 ],
[ 1, 1, -1, -1, -1, 0, 0, 0, 0, 0 ],
[ 1, 0, 0, -1, -1, 0, 0, 0, 0, 0 ],
[ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 1, 0, -1, 0, -1, 0, 0, 0, 0, 0 ],
[ 0, 1, 0, -1, -1, 0, 0, 0, 0, 0 ] ],
[ [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ],
[ 0, 0, 0, 0, 0, 1, 1, -1, -1, -1 ],
[ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ],
[ 1, 1, -1, -1, -1, 0, 0, 0, 0, 0 ] ] ]
</pre></td></tr></table>
<p><a id="X7B3BE908867CE4F9" name="X7B3BE908867CE4F9"></a></p>
<h4>2.3 <span class="Heading">Extension</span></h4>
<p>In this section we introduce some functions for extending a representation of a subgroup to the whole group.</p>
<p><a id="X7950DD2D7B80E583" name="X7950DD2D7B80E583"></a></p>
<h5>2.3-1 ExtendedRepresentation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> ExtendedRepresentation</code>( <var class="Arg">chi, rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Suppose H is a subgroup of a group G and <var class="Arg">chi</var> is an irreducible character of G such that the restriction of <var class="Arg">chi</var> to H, phi say, is irreducible. If <var class="Arg">rep</var> is an irreducible representation of H affording phi then <code class="code">ExtendedRepresentation</code> extends the representation <var class="Arg">rep</var> of H to a representation of G affording <var class="Arg">chi</var>. This function call can be quite expensive when the representation <var class="Arg">rep</var> has a large degree.</p>
<table class="example">
<tr><td><pre>
gap> G := AlternatingGroup( 6 );;
gap> H := Group([ (1,2,3,4,6), (1,4)(5,6) ]);;
gap> chi := Irr( G )[ 2 ];;
gap> phi := RestrictedClassFunction( chi, H );;
gap> IsIrreducibleCharacter( phi );
true
gap> rep := IrreducibleAffordingRepresentation( phi );;
gap> ext := ExtendedRepresentation( chi, rep );
#I Need to extend a representation of degree 5. This may take a while.
[ (1,2,3,4,5), (4,5,6) ] -> [
[ [ 0, 1, 0, -1, -1 ],
[ 0, 0, 0, 1, 0 ],
[ -1, -1, -1, 0, 0 ],
[ 0, 0, 0, 0, -1 ],
[ 0, 0, 1, 1, 1 ] ],
[ [ 1, 0, 1, 0, 1 ],
[ 0, 1, 0, 0, 0 ],
[ -1, -1, 0, 1, 0 ],
[ 1, 1, 1, 0, 0 ],
[ 0, 0, -1, 0, 0 ] ] ]
gap> IsAffordingRepresentation( chi, ext );
true
</pre></td></tr></table>
<p><a id="X83536FA985B4E255" name="X83536FA985B4E255"></a></p>
<h5>2.3-2 ExtendedRepresentationNormal</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> ExtendedRepresentationNormal</code>( <var class="Arg">chi, rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Suppose H is a normal subgroup of a group G and <var class="Arg">chi</var> is an irreducible character of G such that the restriction of <var class="Arg">chi</var> to H, phi say, is irreducible. If <var class="Arg">rep</var> is an irreducible representation of H affording phi then <code class="code">ExtendedRepresentationNormal</code> extends the representation <var class="Arg">rep</var> of H to a representation of G affording <var class="Arg">chi</var>. This function is more efficient than <code class="code">ExtendedRepresentation</code>.</p>
<table class="example">
<tr><td><pre>
gap> G := GL(2,7);;
gap> chi := Irr( G )[ 29 ];;
gap> H := SL(2,7);;
gap> phi := RestrictedClassFunction( chi, H );;
gap> IsIrreducibleCharacter( phi );
true
gap> rep := IrreducibleAffordingRepresentation( phi );;
gap> ext := ExtendedRepresentationNormal( chi, rep );
#I Need to extend a representation of degree 7. This may take a while.
CompositionMapping( [(8,15,22,29,36,43)(9,16,23,30,37,44)
(10,17,24,31,38,45)(11,18,25,32,39,46)(12,19,26,33,40,47)
(13,20,27,34,41,48)(14,21,28,35,42,49),(2,29,12)(3,36,20)
(4,43,28)(5,8,30)(6,15,38)(7,22,46)(9,44,14)(10,16,17)
(11,37,27)(13,23,39)(18,24,25)(19,45,35)(21,31,47)
(26,32,33)(34,40,41)(42,48,49) ] ->
[ [ [ -1, 0, 0, 1, 0, -1, 0 ], [ -1, 0, 0, 0, 0, 0, 0 ],
[ -1, 1, 0, 0, -1, 0, 0 ], [ 0, -1, 0, 0, 0, 0, 0 ],
[ -1, -1, 1, 0, 1, -1, 0 ], [ 0, 0, 0, -1, 0, 0, 0 ],
[ -1, 0, 1, -1, 1, 0, -1 ] ],
[ [ 1, -1, 0, 1, 0, -1, 1 ], [ 1, 0, -1, 1, -1, 0, 1 ],
[ 1, -1, 0, 1, -1, 0, 1 ], [ 0, 0, -1, 0, 0, 0, 0 ],
[ -1, 0, 0, 1, 0, -1, 0 ], [ -1, 0, 0, 0, 0, 0, 0 ],
[ -1, 1, 0, 0, -1, 0, 0 ] ] ], (action isomorphism) )
gap> IsAffordingRepresentation( chi, ext );
true
</pre></td></tr></table>
<p><a id="X80C4BBFE7CDE7AA9" name="X80C4BBFE7CDE7AA9"></a></p>
<h4>2.4 <span class="Heading">Character Subgroups</span></h4>
<p>If chi is an irreducible character of a group G and H is a subgroup of G such that the restriction of chi to H has a linear constituent with multiplicity one, then we call H a character subgroup relative to chi or a chi-subgroup.</p>
<p><a id="X81E72F66831F159E" name="X81E72F66831F159E"></a></p>
<h5>2.4-1 CharacterSubgroupRepresentation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> CharacterSubgroupRepresentation</code>( <var class="Arg">chi</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">> CharacterSubgroupRepresentation</code>( <var class="Arg">chi, H</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a representation affording <var class="Arg">chi</var> by finding a <var class="Arg">chi</var>-subgroup and using the method described in <a href="chapBib.html#biBDix-93">[Dix-93]</a>. If the second argument is a <var class="Arg">chi</var>-subgroup then it returns a representation affording <var class="Arg">chi</var> without searching for a <var class="Arg">chi</var>-subgroup. In this case an error is signalled if no <var class="Arg">chi</var>-subgroup exists.</p>
<p><a id="X79510CED811937D9" name="X79510CED811937D9"></a></p>
<h5>2.4-2 IsCharacterSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> IsCharacterSubgroup</code>( <var class="Arg">chi, H</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>is <code class="code">true</code> if <var class="Arg">H</var> is a <var class="Arg">chi</var>-subgroup and <code class="code">false</code> otherwise.</p>
<table class="example">
<tr><td><pre>
gap> G := AlternatingGroup( 8 );;
gap> chi := Irr( G )[ 2 ];;
gap> H := AlternatingGroup( 3 );;
gap> IsCharacterSubgroup( chi, H );
true
gap> rep := CharacterSubgroupRepresentation( G, chi, H );
[ (1,2,3,4,5,6,7), (6,7,8) ] -> [ [ [
1/3*E(3)+2/3*E(3)^2, 0, 0, -E(3), 0, -1/3*E(3)-2/3*E(3)^2, 1 ],
[ 2/3*E(3)+4/3*E(3)^2, 0, 1, 0, 0, 1/3*E(3)-1/3*E(3)^2, 0 ],
[ 2/3*E(3)+4/3*E(3)^2, 0, 0, 1, 0, 1/3*E(3)-1/3*E(3)^2, 0 ],
[ E(3)^2, 0, 0, 0, 0, 0, 0 ],
[ 2/3*E(3)+4/3*E(3)^2, 0, 0, 0, 1, 1/3*E(3)-1/3*E(3)^2, 0 ],
[ -2/3*E(3)-1/3*E(3)^2, 0, 0, -1, 0, 2/3*E(3)+1/3*E(3)^2, E(3)^2 ],
[ 0, 1, 0, 0, 0, 0, 0 ] ],
[ [ 1, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0 ],
[ 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 0 ],
[ 0, 0, 0, -E(3), E(3), 0, 1 ] ] ]
</pre></td></tr></table>
<p><a id="X8408A559823FED05" name="X8408A559823FED05"></a></p>
<h5>2.4-3 AllCharacterPSubgroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> AllCharacterPSubgroups</code>( <var class="Arg">G, chi</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a list of all p-subgroups of <var class="Arg">G</var> which are <var class="Arg">chi</var>-subgroups. The subgroups are chosen up to conjugacy in <var class="Arg">G</var>.</p>
<p><a id="X86388596828FDAFE" name="X86388596828FDAFE"></a></p>
<h5>2.4-4 AllCharacterStandardSubgroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> AllCharacterStandardSubgroups</code>( <var class="Arg">G, chi</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a list containing well described subgroups of <var class="Arg">G</var> which are <var class="Arg">chi</var>-subgroups. This list may contain Sylow subgroups and their derived subgroups, normalizers and centralzers in <var class="Arg">G</var>.</p>
<p><a id="X7A7B4ECC8722140C" name="X7A7B4ECC8722140C"></a></p>
<h5>2.4-5 AllCharacterSubgroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> AllCharacterSubgroups</code>( <var class="Arg">G, chi</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a list of all <var class="Arg">chi</var>-subgroups of <var class="Arg">G</var> among the lattice of subgroups. This function call can be quite expensive for larger groups. The call is expensive in particular if the lattice of subgroups of the given group is not yet known.</p>
<p><a id="X8459AEDB7B73584F" name="X8459AEDB7B73584F"></a></p>
<h4>2.5 <span class="Heading">Equivalent Representation</span></h4>
<p><a id="X87C25B877C4952BA" name="X87C25B877C4952BA"></a></p>
<h5>2.5-1 EquivalentRepresentation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> EquivalentRepresentation</code>( <var class="Arg">rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>computes an equivalent representation to an irreducible representation <var class="Arg">rep</var> by transforming <var class="Arg">rep</var> to a new basis by spinning up one vector (i.e. getting the other basis vectors as images under the first one under words in the generators). If the input representation, <var class="Arg">rep</var>, is reducible then <code class="code">EquivalentRepresentation</code> does not return any mapping. In this case see section 3.</p>
<table class="example">
<tr><td><pre>
gap> G := SymmetricGroup( 7 );;
gap> chi := Irr( G )[ 2 ];;
gap> rep := CharacterSubgroupRepresentation( G, chi );;
gap> equ := EquivalentRepresentation( rep );
[ (1,2,3,4,5,6,7), (1,2) ] ->
[ [ [ 0, 0, 0, E(5)+E(5)^2+E(5)^3+2*E(5)^4, -1, -E(5)-E(5)^2-E(5)^3-2*E(5)^4 ],
[ E(5)^3-E(5)^4, E(5)^2+E(5)^3+E(5)^4, E(5)+E(5)^3-E(5)^4, -E(5)+E(5)^2
-3*E(5)^3-E(5)^4, -E(5)-E(5)^3+E(5)^4, 2*E(5)-2*E(5)^2+2*E(5)^3 ]
, [ 0, 0, 0, 1, 0, 0 ],
[ 0, 4/5*E(5)+3/5*E(5)^2+2/5*E(5)^3+1/5*E(5)^4, E(5), 1, -E(5),
6/5*E(5)+2/5*E(5)^2+3/5*E(5)^3+4/5*E(5)^4 ], [ 0, 1, 0, 0, 0, 0 ],
[ 0, 0, E(5), 1, -E(5), 2*E(5)+E(5)^2+E(5)^3+E(5)^4 ] ],
[ [ -1, 0, E(5)+E(5)^2+E(5)^3+2*E(5)^4, -E(5)-E(5)^2-3*E(5)^4,
-E(5)-E(5)^2-E(5)^3-2*E(5)^4, E(5)+E(5)^2+3*E(5)^4 ],
[ 0, -1, 0, 0, 0, 0 ],
[ 0, 0, 0, E(5)+E(5)^2+E(5)^3+2*E(5)^4, -1, -E(5)-E(5)^2-E(5)^3-2*E(5)^4
], [ 0, 0, -1, -E(5)^4, 1, E(5)+E(5)^2+E(5)^3+2*E(5)^4 ],
[ 0, 0, -E(5)^4, -E(5)^3+E(5)^4, E(5)+E(5)^2+E(5)^3+2*E(5)^4,
E(5)^3-E(5)^4 ], [ 0, 0, 0, 0, 0, -1 ] ] ]
gap> IsAffordingRepresentation( chi, equ );
true
</pre></td></tr></table>
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