gap-character-tables 1r1p3-5 / usr / share / gap / pkg / ctbllib / doc / ctblothe.tex

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%W  ctblothe.msk          GAP 4 package `ctbllib'               Thomas Breuer
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%H  @(#)$Id: ctblothe.msk,v 1.3 2003/11/19 09:07:50 gap Exp $
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%Y  Copyright (C) 2003,  Lehrstuhl D fuer Mathematik,   RWTH Aachen,  Germany
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\Chapter{Interfaces to Other Data Formats for Character Tables}

This chapter describes data formats for character tables that can be read
or created by {\GAP}.
Currently these are the formats used by
the {\sf CAS} system (see~"Interface to the CAS System"),
the {\MOC} system (see~"Interface to the MOC System"),
and {\GAP}~3 (see~"Interface to GAP 3").




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\Section{Interface to the CAS System}

The interface to {\sf CAS} is thought just for printing the {\sf CAS}
data to a file.
The function `CASString' is available mainly in order to document the
data format.
*Reading* {\sf CAS} tables is not supported; note that the tables
contained in the {\sf CAS} Character Table Library have been migrated to
{\GAP} using a few `sed' scripts and `C' programs.



\>CASString( <tbl> ) F

is a string that encodes the {\sf CAS} library format of the character
table <tbl>.
This string can be printed to a file which then can be read into the
{\sf CAS} system using its `get' command (see~\cite{NPP84}).

The used line length is `SizeScreen()[1]'
(see~"ref:SizeScreen" in the {\GAP} Reference Manual).

Only the known values of the following attributes are used.
`ClassParameters' (for partitions only), `ComputedClassFusions',
`ComputedPowerMaps', `Identifier', `InfoText', `Irr',
`ComputedPrimeBlocks', `ComputedIndicators',
`OrdersClassRepresentatives', `Size', `SizesCentralizers'.



\atindex{CAS tables}{@CAS tables}\atindex{CAS format}{@CAS format}
\atindex{CAS}{@CAS}
\beginexample
gap> Print( CASString( CharacterTable( "Cyclic", 2 ) ), "\n" );
'C2'
00/00/00. 00.00.00.
(2,2,0,2,-1,0)
text:
(#computed using generic character table for cyclic groups#),
order=2,
centralizers:(
2,2
),
reps:(
1,2
),
powermap:2(
1,1
),
characters:
(1,1
,0:0)
(1,-1
,0:0);
/// converted from GAP
\endexample


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\Section{Interface to the MOC System}

The interface to {\MOC} can be used to print {\MOC} input.
Additionally it provides an alternative representation of (virtual)
characters.

The {\MOC}~3 code of a 5 digit number in {\MOC}~2 code is given by the
following list.
(Note that the code must contain only lower case letters.)

\begintt
ABCD    for  0ABCD
a       for  10000
b       for  10001          k       for  20001
c       for  10002          l       for  20002
d       for  10003          m       for  20003
e       for  10004          n       for  20004
f       for  10005          o       for  20005
g       for  10006          p       for  20006
h       for  10007          q       for  20007
i       for  10008          r       for  20008
j       for  10009          s       for  20009
tAB     for  100AB
uAB     for  200AB
vABCD   for  1ABCD
wABCD   for  2ABCD
yABC    for  30ABC
z       for  31000
\endtt

*Note* that any long number in {\MOC}~2 format is divided into packages
of length 4, the first (!) one filled with leading zeros if necessary.
Such a number with decimals $d_1, d_2, \ldots, d_{4n+k}$ is the sequence
$$
0d_1d_2d_3d_4 \ldots 0d_{4n-3}d_{4n-2}d_{4n-1}d_{4n}
   xd_{4n+1}\ldots d_{4n+k}
$$
where $0 \leq k \leq 3$,
the first digit of $x$ is $1$ if the number is positive and $2$ if the
number is negative,
and then follow $(4-k)$ zeros.

\cite{HJLP92} explains details about the {\MOC} system,
a brief description can be found in~\cite{LP91}.



\>MAKElb11( <listofns> ) F

`MAKElb11' prints field information for all number fields with conductor
$n$ where the positive integer $n$ is in the list <listofns>.

The output of `MAKElb11' is used by the {\MOC} system.
`MAKElb11( [ 3 .. 189 ] )' will print something very similar to
Richard Parker's file `lb11'.


\beginexample
gap> MAKElb11( [ 3, 4 ] );
   3   2   0   1   0
   4   2   0   1   0
\endexample

\>MOCTable( <gaptbl> ) F
\>MOCTable( <gaptbl>, <basicset> ) F

`MOCTable' returns the {\MOC} table record of the {\GAP} character table
<gaptbl>.

The first form can be used only if <gaptbl> is an ordinary ($G\.0$) table.
For Brauer ($G\.p$) tables one has to specify a basic set <basicset> of
ordinary irreducibles.
<basicset> must be a list of positions of the basic set characters in the
`Irr' list of the ordinary table of <gaptbl>.

The result is a record that contains the information of <gaptbl>
in a format similar to the {\MOC}~3 format.
This record can, e.g., easily be printed out or be used to print out
characters using `MOCString' (see~"MOCString").

The components of the result are
\beginitems
`identifier' &
    the string `MOCTable(<name>)' where <name> is the `Identifier'
    value of <gaptbl>,

`GAPtbl' &
    <gaptbl>,

`prime' &
    the characteristic of the field (label `30105' in {\MOC}),

`centralizers' &
    centralizer orders for cyclic subgroups (label `30130')

`orders' &
    element orders for cyclic subgroups (label `30140')

`fieldbases' &
    at position $i$ the Parker basis of the number field generated
    by the character values of the $i$-th cyclic subgroup.
    The length of `fieldbases' is equal to the value of label `30110'
    in {\MOC}.

`cycsubgps' &
    `cycsubgps[i] = j' means that class `i' of the {\GAP} table
    belongs to the `j'-th cyclic subgroup of the {\GAP} table,

`repcycsub' &
    `repcycsub[j] = i' means that class `i' of the {\GAP} table
    is the representative of the `j'-th cyclic subgroup of the
    {\GAP} table.
    *Note* that the representatives of {\GAP} table and {\MOC} table
    need not agree!

`galconjinfo' &
    a list $[ r_1, c_1, r_2, c_2, \ldots, r_n, c_n ]$
    which means that the $i$-th class of the {\GAP} table is
    the $c_i$-th conjugate of the representative of
    the $r_i$-th cyclic subgroup on the {\MOC} table.
    (This is used to translate back to {\GAP} format,
    stored under label `30160')

`30170' &
    (power maps) for each cyclic subgroup (except the trivial one)
    and each prime divisor of the representative order store four values,
    namely the number of the subgroup, the power,
    the number of the cyclic subgroup containing the image,
    and the power to which the representative must be raised to yield
    the image class.
    (This is used only to construct the `30230' power map/embedding
    information.)
    In `30170' only a list of lists (one for each cyclic subgroup)
    of all these values is stored, it will not be used by {\GAP}.

`tensinfo' &
    tensor product information, used to compute the coefficients
    of the Parker base for tensor products of characters
    (label `30210' in {\MOC}).
    For a field with vector space basis $(v_1,v_2,\ldots,v_n)$
    the tensor product information of a cyclic subgroup in {\MOC}
    (as computed by `fct') is either 1 (for rational classes)
    or a sequence
$$
n x_{1,1} y_{1,1} z_{1,1} x_{1,2} y_{1,2} z_{1,2}
\ldots x_{1,m_1} y_{1,m_1} z_{1,m_1} 0 x_{2,1} y_{2,1}
z_{2,1} x_{2,2} y_{2,2} z_{2,2} \ldots x_{2,m_2}
y_{2,m_2} z_{2,m_2} 0 \ldots z_{n,m_n} 0
$$
    which means that the coefficient of $v_k$ in the product
$$
\left( \sum_{i=1}^{n} a_i v_i \right) %
\left( \sum_{j=1}^{n} b_j v_j \right)
$$
    is equal to
$$
\sum_{i=1}^{m_k} x_{k,i} a_{y_{k,i}} b_{z_{k,i}}\.
$$
    On a {\MOC} table in {\GAP} the `tensinfo' component is
    a list of lists, each containing exactly the sequence mentioned
    above.

`invmap' &
    inverse map to compute complex conjugate characters,
    label `30220' in {\MOC}.

`powerinfo' &
    field embeddings for $p$-th symmetrizations,
    $p$ a prime integer not larger than the largest element order,
    label `30230' in {\MOC}.

`30900' &
    basic set of restricted ordinary irreducibles in the
    case of nonzero characteristic,
    all ordinary irreducibles otherwise.
\enditems


\>MOCString( <moctbl> ) F
\>MOCString( <moctbl>, <chars> ) F

Let <moctbl> be a {\MOC} table record as returned by `MOCTable'
(see~"MOCTable").
`MOCString' returns a string describing the {\MOC}~3 format of <moctbl>.

If the second argument <chars> is specified, it must be a list of {\MOC}
format characters as returned by `MOCChars' (see~"MOCChars").
In this case, these characters are stored under label `30900'.
If the second argument is missing then the basic set of ordinary
irreducibles is stored under this label.


\beginexample
gap> moca5:= MOCTable( CharacterTable( "A5" ) );
rec( identifier := "MOCTable(A5)", prime := 0, fields := [  ], 
  GAPtbl := CharacterTable( "A5" ), cycsubgps := [ 1, 2, 3, 4, 4 ], 
  repcycsub := [ 1, 2, 3, 4 ], galconjinfo := [ 1, 1, 2, 1, 3, 1, 4, 1, 4, 2 ]
    , centralizers := [ 60, 4, 3, 5 ], orders := [ 1, 2, 3, 5 ], 
  fieldbases := [ CanonicalBasis( Rationals ), CanonicalBasis( Rationals ), 
      CanonicalBasis( Rationals ), 
      Basis( NF(5,[ 1, 4 ]), [ 1, E(5)+E(5)^4 ] ) ], 
  30170 := [ [  ], [ 2, 2, 1, 1 ], [ 3, 3, 1, 1 ], [ 4, 5, 1, 1 ] ], 
  tensinfo := 
    [ [ 1 ], [ 1 ], [ 1 ], [ 2, 1, 1, 1, 1, 2, 2, 0, 1, 1, 2, 1, 2, 1, -1, 2, 
          2, 0 ] ], 
  invmap := [ [ 1, 1, 0 ], [ 1, 2, 0 ], [ 1, 3, 0 ], [ 1, 4, 0, 1, 5, 0 ] ], 
  powerinfo := 
    [ , [ [ 1, 1, 0 ], [ 1, 1, 0 ], [ 1, 3, 0 ], [ 1, 4, -1, 5, 0, -1, 5, 0 ] 
         ], 
      [ [ 1, 1, 0 ], [ 1, 2, 0 ], [ 1, 1, 0 ], [ 1, 4, -1, 5, 0, -1, 5, 0 ] ],
      , [ [ 1, 1, 0 ], [ 1, 2, 0 ], [ 1, 3, 0 ], [ 1, 1, 0, 0 ] ] ], 
  30900 := [ [ 1, 1, 1, 1, 0 ], [ 3, -1, 0, 0, -1 ], [ 3, -1, 0, 1, 1 ], 
      [ 4, 0, 1, -1, 0 ], [ 5, 1, -1, 0, 0 ] ] )
gap> str:= MOCString( moca5 );;
gap> str{[1..70]};
"y100y105ay110fey130t60edfy140bcdfy150bbbfcabbey160bbcbdbebecy170ccbbdd"
gap> moca5mod3:= MOCTable( CharacterTable( "A5" ) mod 3, [ 1 .. 4 ] );;
gap> MOCString( moca5mod3 ){ [ 1 .. 70 ] };
"y100y105dy110edy130t60efy140bcfy150bbfcabbey160bbcbdbdcy170ccbbdfbby21"
\endexample

\>ScanMOC( <list> ) F

returns a record containing the information encoded in the list <list>.
The components of the result are the labels that occur in <list>.
If <list> is in {\MOC}~2 format (10000-format),
the names of components are 30000-numbers;
if it is in {\MOC}~3 format the names of components have `yABC'-format.


\>GAPChars( <tbl>, <mocchars> ) F

Let <tbl> be a character table or a {\MOC} table record,
and <mocchars> either a list of {\MOC} format characters
(as returned by `MOCChars' (see~"MOCChars")
or a list of positive integers such as a record component encoding
characters, in a record produced by `ScanMOC' (see~"ScanMOC").

`GAPChars' returns translations of <mocchars> to {\GAP} character values
lists.


\>MOCChars( <tbl>, <gapchars> ) F

Let <tbl> be a character table or a {\MOC} table record,
and <gapchars> a list of ({\GAP} format) characters.
`MOCChars' returns translations of <gapchars> to {\MOC} format.


\beginexample
gap> scan:= ScanMOC( str );
rec( y105 := [ 0 ], y110 := [ 5, 4 ], y130 := [ 60, 4, 3, 5 ], 
  y140 := [ 1, 2, 3, 5 ], y150 := [ 1, 1, 1, 5, 2, 0, 1, 1, 4 ], 
  y160 := [ 1, 1, 2, 1, 3, 1, 4, 1, 4, 2 ], 
  y170 := [ 2, 2, 1, 1, 3, 3, 1, 1, 4, 5, 1, 1 ], 
  y210 := [ 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 0, 1, 1, 2, 1, 2, 1, -1, 2, 2, 0 ], 
  y220 := [ 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 4, 0, 1, 5, 0 ], 
  y230 := [ 2, 1, 1, 0, 1, 1, 0, 1, 3, 0, 1, 4, -1, 5, 0, -1, 5, 0 ], 
  y050 := [ 5, 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 1, 0, 0 ], 
  y900 := [ 1, 1, 1, 1, 0, 3, -1, 0, 0, -1, 3, -1, 0, 1, 1, 4, 0, 1, -1, 0, 
      5, 1, -1, 0, 0 ] )
gap> gapchars:= GAPChars( moca5, scan.y900 );
[ [ 1, 1, 1, 1, 1 ], [ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ], 
  [ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ], [ 4, 0, 1, -1, -1 ], 
  [ 5, 1, -1, 0, 0 ] ]
gap> mocchars:= MOCChars( moca5, gapchars );
[ [ 1, 1, 1, 1, 0 ], [ 3, -1, 0, 0, -1 ], [ 3, -1, 0, 1, 1 ], 
  [ 4, 0, 1, -1, 0 ], [ 5, 1, -1, 0, 0 ] ]
gap> Concatenation( mocchars ) = scan.y900;
true
\endexample


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Interface to GAP 3}

The following functions are used to read and write character tables in
{\GAP}~3 format.



\>GAP3CharacterTableScan( <string> ) F

Let <string> be a string that contains the output of the {\GAP}~3
function `PrintCharTable'.
In other words, <string> describes a {\GAP} record whose components
define an ordinary character table object in {\GAP}~3.
`GAP3CharacterTableScan' returns the corresponding {\GAP}~4
character table object.

The supported record components are given by the list
`GAP3CharacterTableData'.


\>GAP3CharacterTableString( <tbl> ) F

For an ordinary character table <tbl>, `GAP3CharacterTableString' returns
a string that when read into {\GAP}~3 evaluates to a character table
corresponding to <tbl>.
A similar format is printed by the {\GAP}~3 function `PrintCharTable'.

The supported record components are given by the list
`GAP3CharacterTableData'.


\beginexample
gap> tbl:= CharacterTable( "Alternating", 5 );;
gap> str:= GAP3CharacterTableString( tbl );;
gap> Print( str );
rec(
centralizers := [ 60, 4, 3, 5, 5 ],
fusions := [ rec( name := "Sym(5)", map := [ 1, 3, 4, 7, 7 ] ) ],
identifier := "Alt(5)",
irreducibles := [
[ 1, 1, 1, 1, 1 ],
[ 4, 0, 1, -1, -1 ],
[ 5, 1, -1, 0, 0 ],
[ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ],
[ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ]
],
orders := [ 1, 2, 3, 5, 5 ],
powermap := [ , [ 1, 1, 3, 5, 4 ], [ 1, 2, 1, 5, 4 ], , [ 1, 2, 3, 1, 1 ] ],
size := 60,
text := "computed using generic character table for alternating groups",
operations := CharTableOps )
gap> scan:= GAP3CharacterTableScan( str );
CharacterTable( "Alt(5)" )
gap> TransformingPermutationsCharacterTables( tbl, scan );
rec( columns := (), rows := (), group := Group([ (4,5) ]) )
\endexample

\>`GAP3CharacterTableData' V

This is a list of pairs, the first entry being the name of a component in
a {\GAP}~3 character table and the second entry being the corresponding
attribute name in {\GAP}~4.
The variable is used by `GAP3CharacterTableScan'
(see~"GAP3CharacterTableScan") and `GAP3CharacterTableString'
(see~"GAP3CharacterTableString").




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