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<div class="ChapSects"><a href="chap3.html#X826CC30186DBDB2B">3 <span class="Heading">Reducible Representations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7829A125780DD25D">3.1 <span class="Heading">Constituents of Representations</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7BBA5EC37C52A99D">3.1-1 ConstituentsOfRepresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X87F71650795FE650">3.1-2 IsReducibleRepresentation</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7FF69B0D7DB36D73">3.2 <span class="Heading">Block Representations</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X847B7C45812A563B">3.2-1 EquivalentBlockRepresentation</a></span>
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<h3>3 <span class="Heading">Reducible Representations</span></h3>

<p>In this chapter we introduce some functions which deal with a complex reducible representation <span class="SimpleMath">R</span> of a finite group <span class="SimpleMath">G</span>.</p>

<p><a id="X7829A125780DD25D" name="X7829A125780DD25D"></a></p>

<h4>3.1 <span class="Heading">Constituents of Representations</span></h4>

<p><a id="X7BBA5EC37C52A99D" name="X7BBA5EC37C52A99D"></a></p>

<h5>3.1-1 ConstituentsOfRepresentation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConstituentsOfRepresentation</code>( <var class="Arg">rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>called with a representation <var class="Arg">rep</var> of a group <span class="SimpleMath">G</span>. This function returns a list of irreducible representations of <span class="SimpleMath">G</span> which are constituents of <var class="Arg">rep</var>, and their corresponding multiplicities. For example, if <var class="Arg">rep</var> is a representation of <span class="SimpleMath">G</span> affording a character <span class="SimpleMath">X</span> such that <span class="SimpleMath">X = mY + nZ</span>, where <span class="SimpleMath">Y</span> and <span class="SimpleMath">Z</span> are irreducible characters of <span class="SimpleMath">G</span>, and <span class="SimpleMath">m</span> and <span class="SimpleMath">n</span> are the corresponding multiplicities, then <code class="code">ConstituentsOfRepresentation</code> returns <span class="SimpleMath">[[m, S]</span>, <span class="SimpleMath">[n, T]]</span> where <span class="SimpleMath">S</span> and <span class="SimpleMath">T</span> are irreducible representations of <span class="SimpleMath">G</span> affording <span class="SimpleMath">Y</span> and <span class="SimpleMath">Z</span>, respectively. This function call can be quite expensive when <span class="SimpleMath">G</span> is a large group.</p>

<p><a id="X87F71650795FE650" name="X87F71650795FE650"></a></p>

<h5>3.1-2 IsReducibleRepresentation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsReducibleRepresentation</code>( <var class="Arg">rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>If <var class="Arg">rep</var> is a representation of a group <span class="SimpleMath">G</span> then <code class="code">IsReducibleRepresentation</code> returns <code class="code">true</code> if <var class="Arg">rep</var> is a reducible representation of <span class="SimpleMath">G</span>.</p>

<p><a id="X7FF69B0D7DB36D73" name="X7FF69B0D7DB36D73"></a></p>

<h4>3.2 <span class="Heading">Block Representations</span></h4>

<p><a id="X847B7C45812A563B" name="X847B7C45812A563B"></a></p>

<h5>3.2-1 EquivalentBlockRepresentation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EquivalentBlockRepresentation</code>( <var class="Arg">rep</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">‣ EquivalentBlockRepresentation</code>( <var class="Arg">list</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>If <var class="Arg">rep</var> is a reducible representation of a group <span class="SimpleMath">G</span>, this function returns a block diagonal representation of <span class="SimpleMath">G</span> equivalent to <var class="Arg">rep</var>. If <var class="Arg"> list </var> <span class="SimpleMath">= [[m1, R1]</span>, <span class="SimpleMath">[m2, R2]</span>, ... , <span class="SimpleMath">[mt, Rt]]</span> is a list of irreducible representations <span class="SimpleMath">R1</span>, <span class="SimpleMath">R2</span>, ... , <span class="SimpleMath">Rt</span> of <span class="SimpleMath">G</span> with multiplicities <span class="SimpleMath">m1</span>, <span class="SimpleMath">m2</span>, ... , <span class="SimpleMath">mt</span>, then <code class="code">EquivalentBlockRepresentation</code> returns a block diagonal representation of <span class="SimpleMath">G</span> containing the blocks <span class="SimpleMath">R1</span>, <span class="SimpleMath">R2</span>, ... , <span class="SimpleMath">Rt</span>.</p>

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := AlternatingGroup( 5 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">H := SylowSubgroup( G, 2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">chi := TrivialCharacter( H );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Hrep := IrreducibleAffordingRepresentation( chi );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">rep := InducedSubgroupRepresentation( G, Hrep );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsReducibleRepresentation( rep );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">con := ConstituentsOfRepresentation( rep );</span>
[ [ 1, [ (1,2,3,4,5), (3,4,5) ] -> [ [ [ 1 ] ], [ [ 1 ] ] ] ],
  [ 1, [ (1,2,3,4,5), (3,4,5) ] ->
        [ [ [ E(3), -1/3*E(3)-2/3*E(3)^2, 0, 1/3*E(3)-1/3*E(3)^2 ],
            [ 1, -4/3*E(3)+1/3*E(3)^2, E(3), -2/3*E(3)-1/3*E(3)^2 ],
            [ 1, -E(3), E(3), 0 ],
            [ 1, -1/3*E(3)+1/3*E(3)^2, 1, 1/3*E(3)+2/3*E(3)^2 ] ],
          [ [ 1, -2/3*E(3)-1/3*E(3)^2, 0, 2/3*E(3)+1/3*E(3)^2 ],
            [ 0, -E(3), E(3), 1 ],
            [ 0, -4/3*E(3)-2/3*E(3)^2, E(3), -2/3*E(3)-1/3*E(3)^2 ],
            [ 0, 0, 1, 0 ] ] ] ],
  [ 2, [ (1,2,3,4,5), (3,4,5) ] ->
        [ [ [ -1, 1, 1, 1, -1 ],
            [ 0, 0, 0, 0, 1 ],
            [ -1, 0, 0, 1, -1 ],
            [ 0, 0, 1, 0, 0 ],
            [ 0, -1, 0, -1, 1 ] ],
          [ [ 0, 0, 0, 0, 1 ],
            [ 0, -1, -1, -1, 0 ],
            [ 0, 1, 0, 0, 0 ],
            [ 0, 0, 0, 1, 0 ],
            [ -1, 0, 0, 1, -1 ] ] ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">EquivalentBlockRepresentation( con );</span>
[ (1,2,3,4,5), (3,4,5) ] ->
[ [ [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
    [ 0, E(3), -1/3*E(3)-2/3*E(3)^2, 0, 1/3*E(3)-1/3*E(3)^2, 0,
      0, 0, 0, 0,  0, 0, 0, 0, 0 ],
    [ 0, 1, -4/3*E(3)+1/3*E(3)^2, E(3), -2/3*E(3)-1/3*E(3)^2, 0,
      0, 0, 0, 0, 0, 0, 0, 0, 0 ],
    [ 0, 1, -E(3), E(3), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
    [ 0, 1, -1/3*E(3)+1/3*E(3)^2, 1, 1/3*E(3)+2/3*E(3)^2, 0, 0,
      0, 0, 0, 0, 0, 0, 0, 0 ],
    [ 0, 0, 0, 0, 0, -1, 1, 1, 1, -1, 0, 0, 0, 0, 0 ],
    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ],
    [ 0, 0, 0, 0, 0, -1, 0, 0, 1, -1, 0, 0, 0, 0, 0 ],
    [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ],
    [ 0, 0, 0, 0, 0, 0, -1, 0, -1, 1, 0, 0, 0, 0, 0 ],
    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 1, -1 ],
    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ],
    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, -1 ],
    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ],
    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 1 ] ],
  [ [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
    [ 0, 1, -2/3*E(3)-1/3*E(3)^2, 0, 2/3*E(3)+1/3*E(3)^2, 0, 0,
      0, 0, 0, 0, 0, 0, 0, 0 ],
    [ 0, 0, -E(3), E(3), 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
    [ 0, 0, -4/3*E(3)-2/3*E(3)^2, E(3), -2/3*E(3)-1/3*E(3)^2, 0,
      0, 0, 0, 0, 0, 0, 0, 0, 0 ],
    [ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ],
    [ 0, 0, 0, 0, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0, 0 ],
    [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 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, 1, -1, 0, 0, 0, 0, 0 ],
    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ],
    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, 0 ],
    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ],
    [ 0, 0, 0, 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, 1, -1 ] ] ]
</pre></div>

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