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/a> <a href="chap1_mj.html" >1 </a> <a href="chap2_mj.html" >2 </a> <a href="chap3_mj.html" >3 </a> <a href="chapBib_mj.html" >Bib</a> <a href="chapInd_mj.html" >Ind</a> </div >div class="chlinkprevnexttop" > <a href="chap0_mj.html" >[Top of Book]</a> <a href="chap0_mj.html#contents" >[Contents]</a> <a href="chap2_mj.html" >[Previous Chapter]</a> <a href="chapBib_mj.html" >[Next Chapter]</a> </div >"mathjaxlink" class="pcenter" ><a href="chap3.html" >[MathJax off]</a></p>"X826CC30186DBDB2B" name="X826CC30186DBDB2B" ></a></p>div class="ChapSects" ><a href="chap3_mj.html#X826CC30186DBDB2B" >3 <span class="Heading" >Reducible Representations</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3_mj.html#X7829A125780DD25D" >3 .1 <span class="Heading" >Constituents of Representations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7BBA5EC37C52A99D" >3 .1 -1 ConstituentsOfRepresentation</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X87F71650795FE650" >3 .1 -2 IsReducibleRepresentation</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3_mj.html#X7FF69B0D7DB36D73" >3 .2 <span class="Heading" >Block Representations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X847B7C45812A563B" >3 .2 -1 EquivalentBlockRepresentation</a></span >div ></div >div >3 <span class="Heading" >Reducible Representations</span ></h3>on <span class="SimpleMath" >\(R\)</span > of a finite group <span class="SimpleMath" >\(G\)</span >.</p>"X7829A125780DD25D" name="X7829A125780DD25D" ></a></p>3 .1 <span class="Heading" >Constituents of Representations</span ></h4>"X7BBA5EC37C52A99D" name="X7BBA5EC37C52A99D" ></a></p>3 .1 -1 ConstituentsOfRepresentation</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; ConstituentsOfRepresentation</code >( <var class="Arg" >rep</var td ><td class="tdright" >( function )</td ></tr ></table ></div >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>"X87F71650795FE650" name="X87F71650795FE650" ></a></p>3 .1 -2 IsReducibleRepresentation</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsReducibleRepresentation</code >( <var class="Arg" >rep</var > )</td ><td class="tdright" >( function )</td ></tr ></table ></div >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 "X7FF69B0D7DB36D73" name="X7FF69B0D7DB36D73" ></a></p>3 .2 <span class="Heading" >Block Representations</span ></h4>"X847B7C45812A563B" name="X847B7C45812A563B" ></a></p>3 .2 -1 EquivalentBlockRepresentation</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; 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" >&#an style='color: green'>8227; EquivalentBlockRepresentation</code >( <var class="Arg" >list</var > )</td ><td class="tdright" >( function )</td ></tr ></table ></div >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 >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 ) ] ->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 >div class="chlinkprevnextbot" > <a href="chap0_mj.html" >[Top of Book]</a> <a href="chap0_mj.html#contents" >[Contents]</a> <a href="chap2_mj.html" >[Previous Chapter]</a> <a href="chapBib_mj.html" >[Next Chapter]</a> </div >div class="chlinkbot" ><span class="chlink1" >Goto Chapter: </span ><a href="chap0_mj.html" >Top</a> <a 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