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<title>GAP (SLA) - Chapter 6: Semisimple java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 0
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<meta name=" class="ChapSectshrefc.html#X7FF8A8057E0BFAFD"6<classH"Subalgebras  Lie<span</a>
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<div class="chlinktop"><span span class="ContSS"><br /><span classnbsp&;/><a=chap6java.lang.StringIndexOutOfBoundsException: Range [104, 103) out of bounds for length 131

<div class="chlinkprevnexttop"> <a href="chap0.html">[Topspan>

<p id="mathjaxlink" class="pcenter"><a href="chap6_mj.html">[MathJax on]</a></p>
<p><a id="X7FF8A8057E0BFAFD" name=span class""< < class=">&;nbsp<><a =".X857808A77C13E46C. <a<span>
<div class="ChapSects"><a href="chap6.html#X7FF8A8057E0BFAFD">6 <span class="Heading">Semisimple Subalgebras class=ContSS><br /<span class="nocss">  </span><a href="chap6.html#X82ECC8237E213AAB">6.2-4 InclusionsGraph</a></span>
<div classContSect"<pan class=tocline"><span class="nocss"> </span><a href="chap6.html#X806713FE83F21540">6.1 <span class="Heading"> Branching </span></a>
</span>
<classjava.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 25
<span">6.1-1 ProjectionMatrix<a>/span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X806713FE83F21540"span class"ContSS"</< class""&;nbsp<span>a=c.tml#"6.-10DecompositionOfClosedSet</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7CC8010B7F31B486">6.2 <span class="Heading"> Constructing Semisimple Subalgebras </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href=
<span class="ContSS"><br /><span class
< class"ontSS.tmlX832B17BA819FB057">6.2-3 LieAlgebraAndSubalgebras</a></span>
<span ="ontSS"br /> class="ocss">; </span< hrefchap6html#82ECC8237E213AAB6-4 InclusionsGraph<a<span
<span class="java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0
<span class="<
<span class=ContSS"< /<spanclass="nocss">  </span><a href="chap6.html#X7AAE2B317A4B7D9F">6.2-7 AreLinearlyEquivalentSubalgebras</a
< class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7B1A01FE86A7718F">6.2-8 MakeDatabaseEntry</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X85A6B70B83A36495">6.2-9 ClosedSubsets</a></span>
span=><br /< nocss"nbsp <span>ahref=chap6hX7"6210DecompositionOfClosedSetspan
<;</span><ahref"hap6html#X798DF4D67DE56EB1">6.2-11 IsSpecialClosedSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7D31AA9780B054C1span class="GAPprompt">gap></span> <span class="GAPinput">K:= Subalgebra<pan class=GAPprompt">></span> <span class="GAPinput">L.64, >gap;java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 82
</div></div>
</div>

<h3>6 <span class="Heading">Semisimple Subalgebras of Semisimple Lie Algebras</span></h3>

<p>This chapter contains functions   [ 0, 0, 0, 0, -1, 0, 0 ], [ 0, 0, 000 1, , 0 ,[0,0,0,0,0,-,0 ],

<p><a id="X806713FE83F21540" name="X806713FE83F21540"></a></p>

<h4>6.1 <span class="Heading"> Branching </span><

<

<""java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 270

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ProjectionMatrix</code>( <var class="Arg">L</var>, <var class="Arg">K</var> )</td><td class="tdright">div=> class"width="java.lang.StringIndexOutOfBoundsException: Range [55, 54) out of bounds for length 297
and varclass=""Kvar>are  Liealgebraswith thefollowingproperties<varclass""K<var>is contained <class=Arg></ar,the Cartan subalgebraof<class"rg">L<var>, as  by <var class=Arg>CartanSubalgebra(L)</var> is split (this is automatic if <var class="Arg">L</var> is created by the built in <strong class="pkg">GAP</strong> function) and <var class="Arg">K</var> has a Cartan subalgebra java.lang.StringIndexOutOfBoundsException: Index 479 out of bounds for length 0

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:= SimpleLieAlgebra("E",7,Rationals);;                </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">K:= Subalgebra( L, [ L.1,L.3,L.4,L.5,L.6,L.7,L.63,      [, 0 ,0  ,  , ,0,, ,0,0,]
<span class="GAPprompt">></span> <span class="GAPinput">L.64,L.66,L.67,L.68,L.69,L.70,L.126] );;</span>
<""&;/> span"">K;/pan
63
<span class=&tL algebraof  78  &;
""
<span class=" v.127+(3/2)*v.128)v129()*13052*.131()*.132(2)*133]
[ [ 2, 2, 3, 4< =GAPprompt></> < "APinput  K, [1,,,,,1];java.lang.StringIndexOutOfBoundsException: Index 108 out of bounds for length 108
  [ 0, 0, 0, 0, -1, 0, 0 ], [ 0, 0, 0, 0, 0, -1, 0 ],      0 ,,01,,-/ , 1 0,,  , /
  [ 0, 0, 0, 0, 0, 0, -1 ], [ -1, -2, -2, -3, -2      0, / ,0    ]
</pre></div>

<p><a id="X806713FE83F21540" name="X806713FE83F21540"></a></p>

<h5>6.1-2 Branching</h5>

<div class="      0, 0 0, 0 0 ,3/ ]  ,0 ,0 0,,-/ ,java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Branching</code>( <[ 1,1, 1,1, 1,1 1, 1 ,1,1,,1 ,2] ]
<java.lang.StringIndexOutOfBoundsException: Range [3, 2) out of bounds for length 1139

<div class="example"><pre>
<span class="GAPprompt">gap></span
<span "APprompt">apgt;/><spanclass="GAPinput">K:= Subalgebra( L, [ L.1,L.3,L.4,L.5,L.6,L.7,L.63,               </span>
<span class="GAPprompt">gt;</span><class=GAPinput>.64.66L.7L.L69,., ;<span
<span class="GAPprompt">gap></span> <span java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 0
[ 0, 1 ,[1 1 1,0 0 0,0],[0 ,0,0 1 ,1]java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
      [ 0, 0, 1, 0, "gap&<span>span class=G">();/>
      [ 0, 1, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 0, java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
      [ 0 0 1,0 0, ,0],[1 1, 1 ,1 ,1]]
<span class="GAPprompt">gap></span> <span class="GAPinput">L:= SimpleLieAlgebra("E",7,Rationals);;</span>
<span class=s(L);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">K""
< 78 overRationals&t
<span class="GAPprompt">gap><[[[1,0 0, ,[0, ,0 , 1 ] [1,0,0 ,0]
[ v.127+(3/2)*v.128+(2      [1 0,0, 0,1]  ,,0, 0 ,[0 0,0 ,0 ,java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
<span class="GAPprompt">gap></span> <span class="GAPinput">Branching( L, K, cc, [1,0,0,0,0,0,1] );</span>
[ [ [ 0, 0, 0, 0, 0, 2, -1/2 ], [ 1,<pre>/>
      [ 0, 0, 0, 0, 1, 0, -1/2 ], [ 1, 0, 0, 0, 0, 0, 5/java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      [0 10 ,0 ,1/ ,[2,0 0 ,0 1/2 ,
      [ 0, 0, 1, 0, 0, 0, 1/2 ], [ 0, 0, 0, 0, 0, 1, -5/2 ],
      [1,0,0 , 0,1 32]  1 ,0, 0 0 1,32]
      [ 0, 1, 0, 0, 0, 0, 3/2 ], [ 0, 1, 0, 0, 0, 0, -3/2 ],
      [ 0, 0, 0, 0, 0, 0, 3/2 ], [ 0, 0, java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      [,0 0,0, 0,1 / ,[1,,0 ,0,0 1/  ,
  [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2 ] ]
</<div

<p><a id="X7CC8010B7F31B486" name="X7CC8010B7F31B486"></a></p>

<h4>6.2 <

<p><a id="X7E95AD2C79B19D15" name="X7E95AD2C79B19D15"></a></p>

<h5>6.2-1 h5>6.2-4 InclusionsGraph</h5

<iv"unc"table class=func width"%>tr>td =t><code class>#8227 </code> < =A"L/ )<td< =t"(nbsp;ttribute )/>/><table<div>
<p>Here <var class="Arg">L</var> is a <em>simple</p>Here <var class="Arg">type</var> is a string describing the type of a semisimplejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="example"><pre>
<pan class=GAPpromptgap&t</ spanclass""L=SimpleLieAlgebra(",6R);</span>
<span class=&tLie  dimension28  (3)gt;
<panclass=GAPprompt>><span>< =""LengthK;/java.lang.StringIndexOutOfBoundsException: Range [79, 78) out of bounds for length 79
19
<span class="GAPprompt">gap></span> <span class="GAPinput">K[5];</java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 2
<Lie algebra of dimension 45 over Rationals>
<java.lang.StringIndexOutOfBoundsException: Range [50, 5) out of bounds for length 93
"D5"
<span class="GAPprompt">gap></span> <span class="GAPinput">Branching( L, K[5], [1,0,0,0,0,1] );     </span>
[ [ [ 1, 0, 0java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      [ 1, 0, 0, 0, 1 ], [ 2, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0 ],
      [0,0 0, 0  ,[ ,1,0 0  ,[0 ,0 0,0]]java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
  [ 2, 1, 1, 1, 1, 1, 1, 1, 1 ] ]
<pre><>

<p><a id="X857808A77C13E46C" name="X857808A77C13E46C"></a></p>

<h5>6.2-2 SSSTypes</h5>

<div  theyare notlinearly:
<p>This returns a span class="GAPprompt>/pan span class"APinput"Branching(L sub35] 0,,10 )<span>

<p><a id"X832B17BA819FB057"name"X832B17BA819FB057"</>/p>

<h5>6.2-3 span class="GAPprompt">gap>>  =" ,36,[,,1])<>

" java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 235
<. Example: <var class="Arg">"D5"<var. In generalatype  a sequence simpletypes separatedby spaces Example <var="rg">"2  E6"</var.This function appliccable simple type   in<var class=Arg>/var>hasranklessthan or equal  8 In that case arecord isreturned with two components: <var class="liealg</var>, which is a semisimple Lie algebra of type <var class="Arg">type</var>, and <var class="Arg">subalgs</var> which is the list of its semisimple subalgebras up to linear equivalence. If <var class="Arg">type</var> is a simple type then the list will be simply fetched from the database. Otherwise a computation will be triggered, and afterwards the database will be bigger. (One can check this with a call to <var class="Arg">SSSTypes()</var>). Also we remark that for non-simple types of not so small rank this computation can be difficult.</p>

<p><a id="X82ECC8237E213AAB" name="X82ECC8237E213AAB"></a></p>

<h5>6.2-4 InclusionsGraph</h5>

<div class="func"><table class="func" width="100%">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Here <var class="Arg">type</var> is a string span class="GAPprompt">gap></span> <span class="GAPinput">g:= InclusionsGraph( "C3" );</span>

<div class="example"><pre>
java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 55
<span class=[0  ][,8, ,  ,[9  ,[, , ,
<span class="GAPprompt">gap></span> <span class="GAPinput">L:=[15,12]  0 13], [15,14]  ,  java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
<Lie algebra of dimension 28 over CF(3)>
<span class="GAPprompt">gap></span> <span class="GAPinput">sub:= s.subalgs;;</span>
<java.lang.StringIndexOutOfBoundsException: Range [44, 5) out of bounds for length 81
44
<span class="GAPprompt">gap></span> <span class="GAPinput">span class="GAPprompt">gap></span> <span class="GAPinput">L;/
<spanclass""gap&t<span><spanclass"">[1]<span>
[ 12, 1 ]

# Find the maximal semisimple subalgebraslt;Lie algebraofdimensionover Rationalsgtjava.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
<class="APprompt"gap;/><span class"GAPinput"m= Filtered( g, -&t;x]= ;; i= List m, x -gt; x] );</span>
[ 13, 35, 36, 37, 41, 42, 43, 44 ]
<span class="java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 12
[ "A2", "A1 B2", <p>aid=X82458B807A8D77F6name=X82458B807A8D77F6">/a>/p>

# We see that the subalgebras on positions 35 and 36 are isomorphic;
# however they are not linearly equivalent:
<span class="GAPprompt"<iv ="unc>table class="func""100%"<>td class="tdleft>code class="unc"&8227 DynkinIndex/code><class=Arg"K/var>, varclass="Arg">L</var> )</td><td class="tdright">( operation )</td></tr></table></div>
[ [ [ ,0 1]] [1 ]]
<span class="GAPprompt">gap></span> <span class="GAPinput">Branching( L, sub[36], [0,0,1,0] );</span>
[[[0,1  ,[2 0   ,[,1  java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
</re<div

<p><a span class="GAPprompt">gap></span> <span class="GAPinput">g:= InclusionsGraph( "C7" );;</span>

<h5>6.2-5 SubalgebrasInclusion</h5>

<div class="span class="GAP">ap&;/ < ">: liealg;span>
<p>Here <var class="Arg">K1;algebraof 105 java.lang.StringIndexOutOfBoundsException: Range [48, 47) out of bounds for length 51

< ""<
<span class=" ,"B3, A"" ,B ,"1
<span class="GAPprompt">gap></span> spanjava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 99
[ [ 10, 1 ], [ 11, 1 ], [ ,
  [ 13
  [>java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 62
  [
< java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 140
< java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 112
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 82
<Lie algebra of dimension 21 over Rationals>
<span class="GAPprompt">gap></span> <span class="GAPinput">sub:= s.subalgs;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">K:=SubalgebrasInclusion( L, sub
l;java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 49
<span class=">Here <var classA"<var  a  Lie ,  var "">1/,< ="K2/>java.lang.StringIndexOutOfBoundsException: Range [123, 122) out of bounds for length 586
true
</pre></div>

<p><a id="X82458B807A8D77F6" name="X82458B807A8D77F6"></a></p>

<h5>6.2-6 DynkinIndex</h5>

<div class="func"><table class="funcjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Here <var class="ArgLets  the   the  for C5thatare linearly

<div class="example"><pre>
<class"">gapgt</>< =GAPinput>:=LieAlgebraAndSubalgebras("7 )<span>
<span class="span class="GAPpromptgap&</>span class=GAPinput>:=RegularSemisimpleSubalgebras( L );/>
<spanclass=G">>/>< class=G">: ( g,x-gt; x[]0);i=( ,x-&; [2 )<span>
[63,498,665 804 819 ,822]
<java.lang.StringIndexOutOfBoundsException: Range [22, 5) out of bounds for length 82
<Lie algebra of dimension 105 over Rationals>
spanclass"APprompt"gapg;/pan < class="sub=ssubalgs;/
< class"">&t/>span class"APinput" } java.lang.StringIndexOutOfBoundsException: Range [93, 90) out of bounds for length 103
["" " A ,"6,C ,"2, " C6"]
<span class="GAPprompt">gap></span> <span class="GAPinput">DynkinIndex( sub[665], L );   </span>
[ 7, 4 ]
</pre></div>

<p><a id="X7AAE2B317A4B7D9F" />/java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12

<h5>6

<div class="func"><table class="func" width="100%"><tr>
<Herevar"rg"<> semisimple algebra and<class=><var,< ""><var are. is      <classA"CartanSubalgebra<var)of var class=A"K1,<classA"<>are  the    <class"><var>o var java.lang.StringIndexOutOfBoundsException: Range [380, 379) out of bounds for length 586

<div class="example"><pre>
# Lets find the subalgebras in the database for C5 that are linearly
# equivalent to regular subalgebras:
<span class="GAPprompt">gap></span> <span class="GAPinput">s:= LieAlgebraAndSubalgebras("C5");; L:= s.liealg; sub:= s.subalgs;;</span>
<Lie algebra of dimension 55 over Rationals>
<span class="GAPprompt">gap></span> <span class="GAPinput">reg:= div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AddToDatabase</code>( <var class="Arg">d</var> )</td><td class="pThese  functions that help to savea computedlist of   a semisimple  algebra java.lang.StringIndexOutOfBoundsException: Range [104, 102) out of bounds for length 625
&<>classGjava.lang.StringIndexOutOfBoundsException: Range [80, 60) out of bounds for length 80
<span class="GAPprompt">gap></span> <span class="GAPinput">for K in reg do  </span>
<span class="GAPprompt">></span> <span class="GAPinput">Add(posn,PositionProperty(sub,M -> AreLinearlyEquivalentSubalgebras(L,M,K)));</span>
<span class="GAPprompt">></span> <span class="GAPinput">od;</span>
<span class="GAPprompt
[ 2, 24, 93, 111, 105, 82, 106, 81, 41, 109, 70, 85, 29, 112, 94, 25, 1, 118,
  100, 102, 64, 108, 84, 28, 117, 107, 116, 96, 101, 63, 115, 114, 95, 113 ]
</pre></div>

<p =java.lang.StringIndexOutOfBoundsException: Range [33, 27) out of bounds for length 62

<h5>6.2-8 MakeDatabaseEntry</h5>

<div class="func"><table class="func" width="100%"><< =GAPpromptgapgt;/span spanclass"( A2B2" d="d,;n"<span
<
<p>These

java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 161

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">r:= java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 74
<span class="GAPprompt">gap></span> <span class="GAPinput">:=MakeDatabaseEntry(r );/span>
<span class="GAPprompt">gap></span> <  "D7", "7, "A8,"", "C8" "8, "8, "" ]
</pre></div>

<div class="java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 0
<span class="GAPprompt">gap></span> <span java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 0
<spanclass=GAPprompt"gapgt;/span> span class"GAPinput">AddToDatabase( d);/>
<span class="GAPprompt">gap></span> <span class=<>Herevar="rg><var is a  A  <class"SimpleMath"S/span> of theroots of <var class="rg><var>is to closedifforall <pan class"SimpleMath"ab∈</span  have  <span class=SimpleMath"ab/span>lies  <span SimpleMath"S/>whenever span=""a+/span>isa .Thisfunctioncomputesthelistoftheclosedsubsetsof < class="rg>R/var>up  conjugacy bythe  group. In other words,each closed subset of var class="Arg">R</var> is conjugate under the Weyl group to exactly one element of the output of this function. The output is a list of which each element is a list of roots. A root in such a list, or its negative, lies in the attribute <var class="Arg">PositiveRootsNF( R )</var>.</p>
[ "A1", "A2", "B2", "G2", "A3", "B3", "C3"java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
  "A5", "B5", "C5", "D5", "A6", "B6", "C6", "D6",[1 1,0 0 ] [0,0,1 1], [ 0,1 2,0]  0, 1 ,1],
  "D7", "E7", "A8", "B8", "C8", "D8", "E8", "A2 B2" ]
</pre></div>

<p>a="X85A6B70B83A36495 name="85A6B70B83A36495><a></p>

<h562- ClosedSubsets/h5>

< class"func">table class=func"width="00%>trtd class=tdleft"<code class=func">&'color: green'>8227;ClosedSubsets/>( < =A></ <>< ="&;&bsp)<>/>/>
<Herevar=""<var is  root system  subset<class=SimpleMath"">S/span of  rootsof var class=Arg"Rvarissaidtoiffor <class"impleMath"ab∈S/  havehave that<class"impleMath">+b<span lies in span ="SimpleMath"Sspan wheneverspan class="impleMath">a+b</span> is a root. This function computes the list of the closed subsets of <var class="Arg">R</var> up to conjugacy by the Weyl group. In other words, each closed subset of <var class="Arg">R</var> is conjugate under the Weyl group to exactly one element of the output of this function. The output is a list of which each element java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R:= RootSystem("F",4);</span>
<root system of type
<span class="GAPpromptdiv =""><>
span class"GAPprompt"gap>/span><span class"GAPinput">Lengthc;/>
4844
<span class="GAPprompt">gap></span> <span class="GAPinput">c[1005];</span>
[[1 1 ,0 ],[0,0 1 1]  ,1,2  ,[0, 1,,1 ,
,,,,0, ,2  ,[,1 , ,
   ,,2, 2]  ,2,3  ,[1     ,2 , 2 ]
  [ 1, 2, 4, 2 ], [ 1, 3, 4, 2 ], [ 2, 3, 4, 2 1 ] [ 1 1,1, 1]
</pre></div>

<p><a id="X7A3EAA637F94D973" name="X7A3EAA637F94D973"></a></p>

<h5>6.2-10 DecompositionOfClosedSet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; DecompositionOfClosedSet</code>( <var[1, 3,4,2]  ,3 ,2] ]]
<">c<var ofsomerootsystem havethatvar"rg>c</> is the disjoint union of its symmetric part (consisting of all roots <span class="SimpleMath">a</span> in <var class="Arg">c</var> such that <span class="SimpleMath">-a</span> also lies in <var class="Arg">c</var>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R:= RootSystem("F",4);</span>
<root system of  F4gt;
<span class="GAPprompt">gap></span> <span class="GAPinput">c:= ClosedSubsets(R);;</span>
<span class="GAPprompt">gap></span
[[ ,1,,0 ,[ 0,-,-,0  ,
  [[1,1 0,0 ,  , 0,,1 ,[ 0 ,,1]  ,1 1,1]
      [ 1, 2, &t; system  type F4gjava.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
      [ 1, 2, 3, 1 ], [ 1, 2, 2, 2 ], [ 1, 2, 3, 2 ], [ 1, 2, 4, 2 ],
      [1,3, 4 2, [,3 ,2 ]]]
</pre></div>

<p><a id="X798DF4D67DE56EB1" name="X798DF4D67DE56EB1"></java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

<h5>6.2-11 IsSpecialClosedSet</h5>

<div class="func"><table class
<=">/> is  closed set of roots of some root system. This function returns <var class="Arg">true</var> is <var class="Arg">c</var> is special (that is, for all <span class="SimpleMath">a</span> in <var class="Arg

<div class="example"><pre>
<java.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 91
<root system of type F4>
<span
<span class="GAPprompt">gap>div class=example>pre
false
<span class="GAPprompt">gap&tLalgebra  52over>java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 4
<span classl;root rank4&java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
3579
</pre></div>

<p><a  1 0 ,0][0, ,0  ,[1,0,1]  ,0 2  ,

<h56.212 </h5

< =">table class="funcwidth=100"<>tdclass=tdleft"<>#8227 /(<ar""L> var=""cvar>)<td<classtdright>&;peration)/>tr<table<div
java.lang.StringIndexOutOfBoundsException: Range [64, 60) out of bounds for length 400

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:= SimpleLieAlgebra("F",4,Rationals);</span>
<Lie algebra of dimension 52 over Rationals>
<=GAPprompt>&<span spanclassGAPinput>: ( ;/
<root system of rank 4>
<span class=GAPprompt>g<span spanclass=GAPinput>=ClosedSubsets;;/pan>
<span class="GAPprompt">gap></span> <span class="GAPinput">c[1005];</span>
[ [ 1, 0, 1, 0 ], [ 0, 1, 0, 1 ], [ 1, 0, 1, 1 ], [ 0, 0, 2, 1 ],
  [ 1, 1, 1, 1 ], [ <div
  [ 2, 1, 2, 1 ], [ 1, 1, 3, 2 ], [ 2, 1,ref"hap0html">TopofBook/a&bsp;< href".#contents"[Contents<a&bsp; a href="hap5.tml">Previous Chapter]/a>nbsp; &bsp<href".html">Next ]<a&;  <div>
  [ 2, 1, 4, 2 ], [ 2, 1, 4, 3 ], [ 2, 2, 4, 3 ], [ 0, 0, -2, -1 ] ]
< =GAPprompt><span>span class="GAPinput">K:= ubalgebraOfClosedSetL,c1005])</span
<Lie algebra of dimension 20 over Rationals>
<span class="GAPprompt">gap></span> <span class="java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 0
[ <Lie algebra of dimension 3 over Rationals>,
  <two-sided ideal in <Lie algebra of dimension 20 over Rationals>,
      (imension17& ]
</pre></div>

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