[CTblLib] 4 Interfaces to Other Data Formats for Character Tables

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# 4 Interfaces to Other Data Formats for Character Tables

### Sections

1. Interface to the CAS System
2. Interface to the MOC System
3. Interface to GAP 3

This chapter describes data formats for character tables that can be read or created by GAP. Currently these are the formats used by the 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).

## 4.1 Interface to the CAS System

The interface to CAS is thought just for printing the CAS data to a file. The function `CASString` is available mainly in order to document the data format. Reading CAS tables is not supported; note that the tables contained in the 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 CAS library format of the character table tbl. This string can be printed to a file which then can be read into the CAS system using its `get` command (see NPP84).

  The used line length is `SizeScreen()[1]` (see 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`.

      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

  ## 4.2 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.)

      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

  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 d1, d2, ¼, d4n+k is the sequence

  0d1d2d3d4 ¼0d4n-3d4n-2d4n-1d4n xd4n+1¼d4n+k

  where 0 £ k £ 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.

  HJLP92 explains details about the MOC system, a brief description can be found in 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`.

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

- `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

  `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 [ r1, c1, r2, c2, ¼, rn, cn ] which means that the i-th class of the GAP table is the ci-th conjugate of the representative of the ri-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 (v1,v2,¼,vn) the tensor product information of a cyclic subgroup in MOC (as computed by `fct`) is either 1 (for rational classes) or a sequence

    n x1,1 y1,1 z1,1 x1,2 y1,2 z1,2 ¼x1,m1 y1,m1 z1,m1 0 x2,1 y2,1 z2,1 x2,2 y2,2 z2,2 ¼x2,m2 y2,m2 z2,m2 0 ¼zn,mn 0

    which means that the coefficient of vk in the product

    | æ è | n å i=1 | ai vi | ö ø |  | æ è | n å j=1 | bj vj | ö ø |

    is equal to

    | mk å i=1 | xk,i ayk,i bzk,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.

- `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.

      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"

- `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.

      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

  ## 4.3 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`.

      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) ]) )

- `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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