67.20 Interface to the MOC System

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 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, 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 (v₁,v₂,¼,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 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 v_k 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 prime in [ 2 .. 19 ]; note that the necessary power maps must be stored on gaptbl to compute this component. (label 30230 in MOC)

30900

basic set of restricted ordinary irreducibles in the case of nonzero characteristic, all ordinary irreducibles otherwise.

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 ] ],, 
      [ [ 1, 1, 0 ], [ 1, 2, 0 ], [ 1, 3, 0 ], [ 1, 4, -1, 5, 0, -1, 5, 0 ] ],
      ,,, [ [ 1, 1, 0 ], [ 1, 2, 0 ], [ 1, 3, 0 ], [ 1, 4, 0, 1, 5, 0 ] ],, 
      [ [ 1, 1, 0 ], [ 1, 2, 0 ], [ 1, 3, 0 ], [ 1, 4, -1, 5, 0, -1, 5, 0 ] ],
      ,,, 
      [ [ 1, 1, 0 ], [ 1, 2, 0 ], [ 1, 3, 0 ], [ 1, 4, -1, 5, 0, -1, 5, 0 ] ],
      , [ [ 1, 1, 0 ], [ 1, 2, 0 ], [ 1, 3, 0 ], [ 1, 4, 0, 1, 5, 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"

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.

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.

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 := [ 19, 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 4, 0, 1, 5, 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

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