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Quellcode-Bibliothek o8p2s3_o8p5s3.xml Sprache: XML

<

<<!- W  o8p2s3_o8p5s3.java.lang.StringIndexOutOfBoundsException: Range [31, 26) out of bounds for length 77
<& ComputationsM>_+5._</M> M./><>

Date: October 08th, 2006

<P/>

This chapter shows how to construct a representation of the automorphic
extension <M>G</M> of the simple group <M>S = O_8^+(5)</M>
by a symmetric group on three points,
together with an embedding of
!-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%-java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77

P>

As an application, it is shown that the permutation representation of <M>G</M>
n  cosetsof <H/  a baseoflengthjava.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
java.lang.StringIndexOutOfBoundsException: Range [9, 4) out of bounds for length 48

<-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ->
< Label":verview_o8p2s3java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
<eading>verview/eading

Let MO8+)/ such that the normalizer <M>H</M>, say,
<M>O_8^+(5) \cong </of <M><Min<><M>is  automorphic  ofMM/>
that is,the  simplegroupthat occursas  composition
of the general orthogonalMO8+2<M>java.lang.StringIndexOutOfBoundsException: Range [17, 18) out of bounds for length 17
matrices over the field with five elements.

<P/>

The outer automorphism group of <M>S</M> is Analogously, let <M>M.2 = H \cap S.2</M> and <M>M.3 = H \cap S.3</M>.
on four pointsjava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
Let<G/>beautomorphicextension of <M>S</M> by the symmetric group
on three points.
By <Cite Key="Kle87"/>,
the group <M>S</M> contains a maximal subgroup <M>M</M> of theand the reduction of its rational <M>8</M>-dimensional representation modulo
<<M>5/M> embeds into the general orthogonal group GO<M>{}^+(8,5)</M>,
f<><M  <><M  anautomorphicextension of <></>by
a Thenthe of GO<>}+8)/M>   isomorphicimage  <><M
(GO>{^(5<M>on <><M>spaces  in the
<>^+(2)/>)

<P/>

Let <M>S.2</M> and <M>S.3</M> denote intermediate java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 29
<M>S</M> and <M>G</M>,
java.lang.StringIndexOutOfBoundsException: Range [7, 2) out of bounds for length 70
Analogously, let <M>M.2 = H \cap S.2</M> java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<P/>

In Section <Ref Subsect="sect:sect2"/>,
we use the followingwe take  outer M\</>of MM<>  orderthree,
<S2/.
By;Cjava.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 42
java.lang.StringIndexOutOfBoundsException: Range [20, 3) out of bounds for length 75
the of its rational<M><M- representationmodulo
<M>5</M> embeds into the general orthogonal group GO<M>{}^+(yieldspermutationrepresentations  MS.</>and its subgroup
which has the structure <M>2.O_8^+(5).2^2</
java.lang.StringIndexOutOfBoundsException: Range [35, 4) out of bounds for length 77
in GO<M>{}^+(8,5)</Mw put theabove  together,
natural module of <>^(8,5)<>yield<>.2<M a subgroupof
(a supergrouppoints.
where both groups are represented as permutation groups on <M>N = 19\,656</M>
java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 7

<P/>

In Section <Ref Sect="sect:sect3"/>,
first we use &has a base of length two;
Then we take an outer automorphism <M>\alpha<this question arose &bsp< Key"">java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
and extend f   characterM(1HG)_/>
,we  the images of generating  of <</>and <M>M</M>
<>alpha/>and<\^<M>
This yields permutation see Section <Ref Subsect="sect:appendix_data"/>).
<M>M.3<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<P/>

In Section <Ref Sect="sect:sect4"S Label=sect:java.lang.StringIndexOutOfBoundsException: Range [28, 26) out of bounds for length 28
we  the above informationtogether
in order to construct permutation representations of <M>G</M> and
on M3N<M>points.

<P/>

As an application, it is shown in Section Heading> MatrixRepresentation the   ofTypeE8/>/eading
thatinn;CiteKey="CN85 =p.85,Section">
has a base of we can generate the Weyl group <M>W</M> of type <M>E_8</M> as a group of
this questionarose in; Key=B">java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48

</

In two appendices, it plus the vectors obtained from these by permuting the coordinates,
from the permutation character<(_^G)_<M>
(see Section&
java.lang.StringIndexOutOfBoundsException: Range [10, 3) out of bounds for length 72
(   \(\m 1/,\m /,p 1/,\m 1/2,

</Section>


<Subsection Label="subsect:120pt">
<Heading>Compatible Generators of <M>M</M>, <M>M.2</M>, <M>S</M>, and <M>S.2</M></Heading>

For the next computations,
we switch from the natural matrix representation of GO<M>{}^+(8,5)</M>
to a permutation representation of PGO<M>{}^+(8,5)</M>,
of degree <M>N = 19\,656</M>,
which is given by the action of GO<M>{}^+(8,5)</M> on the smallest orbit
of <M>1</M>-spaces in its natural module.

<P/>

<Example><![CDATA[
gap> orbs:= OrbitsDomain( largegroup, NormedRowVectors( GF(5)^8 ),
>                         OnLines );;
gap> List( orbs, Length );
[ 39000, 39000, 19656 ]
gap> N:= Length( orbs[3] );
19656
gap> orbN:= SortedList( orbs[3] );;
gap> largepermgroup:= Action( largegroup, orbN, OnLines );;
]]></Example>

<P/>

In the same way,
permutation representations of the subgroup <M>M.2 \cong </M>SO<M>{}^+(8,2)</M>
and of its derived subgroup <M>M</M> are obtained.
But first we compute a smaller generating set of the simple group <M>M</M>,
using a permutation using a permutation representation on <M>120</M> points.

<P/>

<Example><![CDATA[
gap> orbwe8:= SortedList( Orbit( we8, rootvectors[1], OnLines ) );;
gap> Length( orbwe8 );
120
gap> java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 4
gap> m2_120:= Imagegap (orbwe8 ;
p> m_120:= DerivedSubgroup(m2_120 );;
Length );
2
gap> gens_m:= List( sml, x -> PreImagesRepresentative( we8_to_m2, x ) );;
]><Example

<P/>

Now we compute the actions of <M>Mg>sml:=SmallGeneratingSetjava.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 56
on the above orbitjava.lang.StringIndexOutOfBoundsException: Range [0, 19) out of bounds for length 13
generating<>.<M> choose  <bN iM.2\java.lang.StringIndexOutOfBoundsException: Range [73, 72) out of bounds for length 79
which is obtained action  matrix <M>b \in 2.M.2 \setminus 2.M</M>.

P/

<Example><![CDATA[
gap> gens_m_N
>x- (  x*I^1 ,OnLines));
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
x-(I*   ^-1,orbNO );java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
>DeterminantMatb ;
Z(5)^2
gap> b_N:= Permutation( b, orbN, OnLines );;
gap> m2_Ngap DeterminantMat ;
]]>/>

<P/>

(Note that <M>M.2</M> is not contained in PSO<M>{}^+(8,5)</M>,
since the java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<P/>

java.lang.StringIndexOutOfBoundsException: Range [18, 3) out of bounds for length 66
and <M>S.2</M> isjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<P/>

<Example><![CDATA[
gap> s_N:= and <M>S.2</M> is generated by <M>S</M> together with <M>b_N</M>.
gap> java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 0
]>/Example

</Subsection>

</Section>



</Subsection>


<Subsection Label="subsect:M.3andS.3">
<Heading>The Action of <M>S.3</M> on <M>S</M></Heading>

Our approach is to extend the automorphism <M>\alpha</M> of <M>M</M> to
<M>S</M>;
we can do this because in the full automorphism group of <M>S</M>,
<E>any</E> <M>O^+_8(2)</M> type subgroup extends
to a group of the type <M>O^+_8(2).3</M>, and this extension lies in a
subgroup of the type <M>O^+_8(5).3</M> (see<M> has a root <M>s</M> of order

<P/>

The group <M>M</M> is maximal in <M>S</M>,
so <M>S</M> is generated by <M>M</and <M>C_S(x)</M> has order <M>189<
<M>s \in S  element <>y \n C_S(M)\)/>has  M><M>java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
Having fixed such an element equalto theof   of My/>and imagesof these
java.lang.StringIndexOutOfBoundsException: Range [10, 2) out of bounds for length 65
2/.

<>

For that, we first choose <M>x \in M</M> such that <M>C_S(xjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
>    gens_m_N,gens_m_N_alpha;
Then we choose <M>s \in C_S(x) \setminus M</M>,
java.lang.StringIndexOutOfBoundsException: Range [20, 3) out of bounds for length 69
we then check returnOrder ) (cand) and
image.

<P/>

Each element <M>x</M>   java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
M63/>in<M><M>
and <M>      until( )=63;
For such <>x/>
      good:= Filtered( Elements( Centralizer( s_N, c_m_alpha ) ),
satisfies the necessary x - ( gens_m_N,
that the orders of >gens_m_N_alpha,s,x ))
are equal to the orders >    until Length( good )
generators under <M>\gap> c_m_alpha_2:= Images( alpha, c_m_alpha );;
In other words, we have <M>gap> = Filtered Elements (  c_m_alpha_2))java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66

<P/>

<Example><![CDATA[
gap> : [];
>]]>/Example
gap> CheapTestForHomomorphism:= java.lang.StringIndexOutOfBoundsException: Range [1, 40) out of bounds for length 4
>        returnthis meansthat   of<>3/ on M3<>
>               ForAll( [ 1 .. Length( gens ) ],
>               [i] * x )=Ordergenimages[]   ))java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
> end;;
gap> \left \egin{rray}{cc}
t
>        & \ amp;\java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
>      until Order(           \end{array}],
>      java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 6
>      repeat
>        s:= Random( c_s );java.lang.StringIndexOutOfBoundsException: Range [70, 24) out of bounds for length 70
>      until Order
>      :  ,Centralizerm_N, ;
>      good:= Filtered( Elements( Centralizer( s_N, c_m_alpha ) ),
>               x java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
>                      gens_m_N_alpha, s, x ) );
>    until Length( good ) =]</>
gap> s_alpha:= good[1]P>
gapjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gap> good:= Filtered( Elements( which yields generators for <M>S</M> that are compatible with those of
>      <M>M</M>
>                                     s_alpha, x )<>
gap> s_alpha_2
]>/>

<P/>

Using notation  previoussection,
thisjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
be extendedMS.<>by   permutation corresponding
to the block diagonal java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 0
<lt Only="aTeX>
\[
  l begina}ccc}
            s     &    & \\
            &<>  Generators of <M>H</M> and <M>G</M></Heading>
            & &After having compatiblerepresentations MM2/>
          \end{array} \right] ,
\]
</Alt>

<Alt Only="Text">
<Verb>
                         ⌈ s             ⌉
                          sα|
                     (^)⌋java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
</Verb>
</Alt>

<Alt Only="HTML"><![java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 43
<ivclass=p">
    Theactiononfirst M></M   thatofthe
    <tr>
java.lang.StringIndexOutOfBoundsException: Range [33, 6) out of bounds for length 64
      <td class=" Sectionn;RefSubsect="ubsect:120pt".
      <td class="tdright"><span class="SimpleMath"> </span></td>
    </List
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      <td class=Usingthe  the  ,
td class="tdright"><span ^s><td>
      <td class="tdright"><span class="SimpleMath"> </span></td>
    </tr>
    <tr><ltOnly"aTeX>
      <td class="tdright"><span class="SimpleMath"> </span></td>
      <td class="left[ \begin{array}{ccc}
      <td class="tdright"><spanb     && \
    </tr>
  </table>
</div>
]]></Alt>

as\end{rray right ,

<P/>

<Example><![<lt Only=""
gap<Verb>
gap⌈
]

<>

(And of java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 8
hichyieldsfor <><M> that are  with thoseof
<M>M</M>.)

<P/>

<Example><![CDATA[
gap> s_3N:= ClosureGroup( m_3N, java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 9
]]></Example>

<Subsection>

</Section>



<P/into<><M   java.lang.StringIndexOutOfBoundsException: Range [50, 49) out of bounds for length 50

Now we consider the restrictionsin decomposition.
<M><M><>.2/M> MM3/>
Note that
<M>(1_H^G)_M = (1_M^S)_M</M>,
M(_G{M.2}=(_M.}{.)M2<M>
<M>(1_H^Gwhich   than<>S:]/>

<P/>

<java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 18
gap> tblm2:= CharacterTable( "O8+(2)
ap : CharacterTable "8+2.");
gap> tblm:= CharacterTable( "O8+(2)" );;
gap> pi_m2:= pi{ GetFusionMap( java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 11
gap> pi_m3:= pi{ GetFusionMap( tblm3, tblh )
gap> pi_m:= pi_m3{ GetFusionMap( tblm, tblm3 ) };;
]>/>

<P/>

character<>1_^S_M<M>decomposes
into <M>483on <>/<M>
and  MM/>orbits on >S/> to regular constituents
in this M>1_{\langle g \rangl/  <M>_^_<M atjava.lang.StringIndexOutOfBoundsException: Range [62, 61) out of bounds for length 74
If there is no regular transitive constituent in <M>(1
the largest degree of a transitive constituent is <M>|M|/2</in<>/>.java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
but then the degree of <M>1_M^S</M>
whichsmallerthan >[:]M>.

<P/>

<Example><![CDATA[
gap> n:=  inv:= Filtered .( tblm2 )]java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
483
>                     i in ClassPositionsOfDerivedSubgroup(tblm2 ) )java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
42065049600
gap> pi[1];
51162109375
]]></Example>

<P/>

For the case of <M>M.2 < S.2</M>, this argument turns out to be not
sufficient.
So we first compute a lower bound on the number of java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 0
on <M>S/MList( java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
For involutions <M>g \in M</M ,15]
M1_{\angle  r}^<M>  M(_^S_M<M> is atmost integral
part
<M>1_gap>69 (  ) + - -1)*Size    +1;
from this we compute that there are at most <M>208</M> such constituents.

<P/

<Example>P>
gap> inv:= Filtered( [ 1 .. URL>http://www.math.rwth-aachen.de/~Thomas.Breuer/ctbllib/data/o8p2s3_o8p5s3.g</URL>
>              i ->P/>
[ 2, 3, 4, 5, 6 ]
gap> n2:= List( inv,
>           - Int(2 [i] / SizesCentralizers )i]))java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
[ 1, 54, 54, 54, java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 65
gap ( n2)java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
208
]]></Example>

<P/>

As ajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<M>S/M</M>.

<>

<Example><![CDATA[

>      i -> i * Size( tblm ) + 208 * Size( tblm ) / 2
>           + ( 483 - i - 208 - 1 ) * Size( tblm ) / 3 + 1 >= pi[1] )restricted <><>
148
]]></Example>

<P/>

Now we consider the action of <M>M.2</M> on <M>S.2/M.2</M>.
If <M>M.2</M> has no regular orbit then the <M>148</M> regular orbits of
<M>M</M> must arise from the restriction of transitive constituents
<M>1_U^{M.2}</M> to <M>    <Ref =Identifier" BookName=BookName="ref"/>
and such that <M>U</M> is not contained in <M>M</M>.
(This follows from the fact that the restriction of a transitive constituent
of <M>(1_{M.2}^{S.2})_{M.2}<<><M is   a
constituent of <M>(1_M^S)_M</M> or the sum of two such constituents;
the latter case occurs if and only if the point stabilizer is contained
in <M>M</M>.)
However, the number of these constituents is at most <M>134</M>.

<P/>

<Example><![CDATA[
gap> inv:= Filtered( [ 1 .. NrConjugacyClasses( tblm2 ) ],
>              i -> OrdersClassRepresentatives( tblm2 )[i] = 2 and
>                   not i in ClassPositionsOfDerivedSubgroup( tblm2 ) );
[ 41, 42 ]
gap>n2=List(inv,
>           i -> java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 7
[ 108, 26 ]
gap> Sum( n2 );
134
]]></Example>



<P/>

Finally, we consider the action/Item
We compute that <M<Mark>Cjava.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 26
transitiveconstituents
and at most <M>69</M>     permutations of degr of degree<></M>
This is already sufficient to show that there java.lang.StringIndexOutOfBoundsException: Range [0, 50) out of bounds for length 26

<<langle   rjava.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48

<Example><![CDATA[
gap> n:= ScalarProduct( tblm3, pi_m3, TrivialCharacter( tblm3 ) );<tem><>cong</<
05
gap> inv:= Filtered(<>
>R>
[ 2, 3, 4 ]
gap> n2:= List( inv,
>           i -> Int( 2 * pi_m3[i] / java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 5
[ 0, 54, 15 ]
gap   )
69
gap> 69 * Size( tblm3 ) / 2 + ( java.lang.StringIndexOutOfBoundsException: Range [5, 6) out of bounds for length 5
41542502401
pi[;
51162109375
]]></Example>

<><M regular constituents->

</Section>

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