  
  [1X6 [33X[0;0YInterfaces to Other Data Formats for Character Tables[133X[101X
  
  [33X[0;0YThis chapter describes data formats for character tables that can be read or
  created by [5XGAP[105X. Currently these are the formats used by[133X
  
  [30X    [33X[0;6Ythe [5XCAS[105X system (see [14X6.1[114X),[133X
  
  [30X    [33X[0;6Ythe [5XMOC[105X system (see [14X6.2[114X),[133X
  
  [30X    [33X[0;6Y[5XGAP[105X 3 (see [14X6.3[114X),[133X
  
  [30X    [33X[0;6Ythe so-called [13XCambridge format[113X (see [14X6.4[114X), and[133X
  
  [30X    [33X[0;6Ythe [5XMAGMA[105X system (see [14X6.5[114X).[133X
  
  
  [1X6.1 [33X[0;0YInterface to the [5XCAS[105X[101X[1X System[133X[101X
  
  [33X[0;0YThe interface to [5XCAS[105X (see [NPP84]) is thought just for printing the [5XCAS[105X data
  to  a  file.  The function [2XCASString[102X ([14X6.1-1[114X) is available mainly in order to
  document the data format. [13XReading[113X [5XCAS[105X tables is not supported; note that the
  tables  contained  in  the [5XCAS[105X Character Table Library have been migrated to
  [5XGAP[105X using a few [10Xsed[110X scripts and [10XC[110X programs.[133X
  
  [1X6.1-1 CASString[101X
  
  [29X[2XCASString[102X( [3Xtbl[103X ) [32X function
  
  [33X[0;0Yis  a string that encodes the [5XCAS[105X library format of the character table [3Xtbl[103X.
  This  string  can  be  printed to a file which then can be read into the [5XCAS[105X
  system using its [10Xget[110X command (see [NPP84]).[133X
  
  [33X[0;0YThe  used  line length is the first entry in the list returned by [2XSizeScreen[102X
  ([14XReference: SizeScreen[114X).[133X
  
  [33X[0;0YOnly  the known values of the following attributes are used. [2XClassParameters[102X
  ([14XReference:  ClassParameters[114X)  (for  partitions  only), [2XComputedClassFusions[102X
  ([14XReference:     ComputedClassFusions[114X),     [2XComputedIndicators[102X    ([14XReference:
  ComputedIndicators[114X),   [2XComputedPowerMaps[102X   ([14XReference:   ComputedPowerMaps[114X),
  [2XComputedPrimeBlocks[102X     ([14XReference:     ComputedPrimeBlockss[114X),    [2XIdentifier[102X
  ([14XReference:   Identifier   (for  character  tables)[114X),  [2XInfoText[102X  ([14XReference:
  InfoText[114X),  [2XIrr[102X  ([14XReference:  Irr[114X),  [2XOrdersClassRepresentatives[102X  ([14XReference:
  OrdersClassRepresentatives[114X),   [2XSize[102X   ([14XReference:  Size[114X),  [2XSizesCentralizers[102X
  ([14XReference: SizesCentralizers[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPrint( CASString( CharacterTable( "Cyclic", 2 ) ), "\n" );[127X[104X
    [4X[28X'C2'[128X[104X
    [4X[28X00/00/00. 00.00.00.[128X[104X
    [4X[28X(2,2,0,2,-1,0)[128X[104X
    [4X[28Xtext:[128X[104X
    [4X[28X(#computed using generic character table for cyclic groups#),[128X[104X
    [4X[28Xorder=2,[128X[104X
    [4X[28Xcentralizers:([128X[104X
    [4X[28X2,2[128X[104X
    [4X[28X),[128X[104X
    [4X[28Xreps:([128X[104X
    [4X[28X1,2[128X[104X
    [4X[28X),[128X[104X
    [4X[28Xpowermap:2([128X[104X
    [4X[28X1,1[128X[104X
    [4X[28X),[128X[104X
    [4X[28Xcharacters:[128X[104X
    [4X[28X(1,1[128X[104X
    [4X[28X,0:0)[128X[104X
    [4X[28X(1,-1[128X[104X
    [4X[28X,0:0);[128X[104X
    [4X[28X/// converted from GAP[128X[104X
  [4X[32X[104X
  
  
  [1X6.2 [33X[0;0YInterface to the [5XMOC[105X[101X[1X System[133X[101X
  
  [33X[0;0YThe  interface  to  [5XMOC[105X  (see [HJLP])  can  be  used  to  print  [5XMOC[105X  input.
  Additionally   it   provides  an  alternative  representation  of  (virtual)
  characters.[133X
  
  [33X[0;0YThe  [5XMOC[105X 3  code of a [22X5[122X digit number in [5XMOC[105X 2 code is given by the following
  list. (Note that the code must contain only lower case letters.)[133X
  
  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
  
  [33X[0;0Y[13XNote[113X that any long number in [5XMOC[105X 2 format is divided into packages of length
  [22X4[122X,  the  first (!) one filled with leading zeros if necessary. Such a number
  with  decimals [22Xd_1, d_2, ..., d_{4n+k}[122X is the sequence [22X0 d_1 d_2 d_3 d_4 ...
  0 d_{4n-3} d_{4n-2} d_{4n-1} d_4n d_{4n+1} ... d_{4n+k}[122X where [22X0 ≤ k ≤ 3[122X, the
  first  digit  of  [22Xx[122X  is  [22X1[122X  if the number is positive and [22X2[122X if the number is
  negative, and then follow [22X(4-k)[122X zeros.[133X
  
  [33X[0;0YDetails  about  the  [5XMOC[105X system are explained in [HJLP], a brief description
  can be found in [LP91].[133X
  
  [1X6.2-1 MAKElb11[101X
  
  [29X[2XMAKElb11[102X( [3Xlistofns[103X ) [32X function
  
  [33X[0;0YFor  a list [3Xlistofns[103X of positive integers, [2XMAKElb11[102X prints field information
  for all number fields with conductor in this list.[133X
  
  [33X[0;0YThe  output  of [2XMAKElb11[102X is used by the [5XMOC[105X system; Calling [10XMAKElb11( [ 3 ..
  189 ] )[110X will print something very similar to Richard Parker's file [11Xlb11[111X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMAKElb11( [ 3, 4 ] );[127X[104X
    [4X[28X   3   2   0   1   0[128X[104X
    [4X[28X   4   2   0   1   0[128X[104X
  [4X[32X[104X
  
  [1X6.2-2 MOCTable[101X
  
  [29X[2XMOCTable[102X( [3Xgaptbl[103X[, [3Xbasicset[103X] ) [32X function
  
  [33X[0;0Y[2XMOCTable[102X returns the [5XMOC[105X table record of the [5XGAP[105X character table [3Xgaptbl[103X.[133X
  
  [33X[0;0YThe  one  argument  version  can be used only if [3Xgaptbl[103X is an ordinary ([22XG.0[122X)
  table.  For  Brauer ([22XG.p[122X) tables, one has to specify a basic set [3Xbasicset[103X of
  ordinary  irreducibles.  [3Xbasicset[103X  must  then  be a list of positions of the
  basic  set characters in the [2XIrr[102X ([14XReference: Irr[114X) list of the ordinary table
  of [3Xgaptbl[103X.[133X
  
  [33X[0;0YThe  result  is a record that contains the information of [3Xgaptbl[103X in a format
  similar  to  the [5XMOC[105X 3 format. This record can, e. g., easily be printed out
  or be used to print out characters using [2XMOCString[102X ([14X6.2-3[114X).[133X
  
  [33X[0;0YThe components of the result are[133X
  
  [8X[10Xidentifier[110X[108X
        [33X[0;6Ythe  string  [10XMOCTable( [110X[22Xname[122X[10X )[110X where [22Xname[122X is the [2XIdentifier[102X ([14XReference:
        Identifier (for character tables)[114X) value of [3Xgaptbl[103X,[133X
  
  [8X[10XGAPtbl[110X[108X
        [33X[0;6Y[3Xgaptbl[103X,[133X
  
  [8X[10Xprime[110X[108X
        [33X[0;6Ythe characteristic of the field (label [10X30105[110X in [5XMOC[105X),[133X
  
  [8X[10Xcentralizers[110X[108X
        [33X[0;6Ycentralizer orders for cyclic subgroups (label [10X30130[110X)[133X
  
  [8X[10Xorders[110X[108X
        [33X[0;6Yelement orders for cyclic subgroups (label [10X30140[110X)[133X
  
  [8X[10Xfieldbases[110X[108X
        [33X[0;6Yat  position  [22Xi[122X  the Parker basis of the number field generated by the
        character values of the [22Xi[122X-th cyclic subgroup. The length of [10Xfieldbases[110X
        is equal to the value of label [10X30110[110X in [5XMOC[105X.[133X
  
  [8X[10Xcycsubgps[110X[108X
        [33X[0;6Y[10Xcycsubgps[i]  =  j[110X  means that class [10Xi[110X of the [5XGAP[105X table belongs to the
        [10Xj[110X-th cyclic subgroup of the [5XGAP[105X table,[133X
  
  [8X[10Xrepcycsub[110X[108X
        [33X[0;6Y[10Xrepcycsub[j]  =  i[110X  means  that  class  [10Xi[110X  of  the  [5XGAP[105X  table  is the
        representative of the [10Xj[110X-th cyclic subgroup of the [5XGAP[105X table. [13XNote[113X that
        the representatives of [5XGAP[105X table and [5XMOC[105X table need not agree![133X
  
  [8X[10Xgalconjinfo[110X[108X
        [33X[0;6Ya list [22X[ r_1, c_1, r_2, c_2, ..., r_n, c_n ][122X which means that the [22Xi[122X-th
        class  of  the [5XGAP[105X table is the [22Xc_i[122X-th conjugate of the representative
        of  the  [22Xr_i[122X-th  cyclic  subgroup  on  the [5XMOC[105X table. (This is used to
        translate back to [5XGAP[105X format, stored under label [10X30160[110X)[133X
  
  [8X[10X30170[110X[108X
        [33X[0;6Y(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 [10X30230[110X power map/embedding information.) In [10X30170[110X
        only  a  list  of  lists  (one  for each cyclic subgroup) of all these
        values is stored, it will not be used by [5XGAP[105X.[133X
  
  [8X[10Xtensinfo[110X[108X
        [33X[0;6Ytensor  product  information,  used to compute the coefficients of the
        Parker  base  for  tensor products of characters (label [10X30210[110X in [5XMOC[105X).
        For  a  field with vector space basis [22X(v_1, v_2, ..., v_n)[122X, the tensor
        product  information  of a cyclic subgroup in [5XMOC[105X (as computed by [10Xfct[110X)
        is either [22X1[122X (for rational classes) or a sequence[133X
  
  
              [33X[1;12Y[24X[33X[0;6Yn x_1,1 y_1,1 z_1,1 x_1,2 y_1,2 z_1,2 ... 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 ... x_2,m_2
              y_2,m_2 z_2,m_2 0 ... z_n,m_n 0[133X[124X[133X
  
  
        [33X[0;6Ywhich means that the coefficient of [22Xv_k[122X in the product[133X
  
  
              [33X[1;12Y[24X[33X[0;6Y( ∑_i=1^n a_i v_i ) ( ∑_j=1^n b_j v_j )[133X[124X[133X
  
  
        [33X[0;6Yis equal to[133X
  
  
              [33X[1;12Y[24X[33X[0;6Y∑_i=1^m_k x_k,i a_y_k,i} b_z_k,i} .[133X[124X[133X
  
  
        [33X[0;6YOn a [5XMOC[105X table in [5XGAP[105X, the [10Xtensinfo[110X component is a list of lists, each
        containing exactly the sequence mentioned above.[133X
  
  [8X[10Xinvmap[110X[108X
        [33X[0;6Yinverse  map  to  compute complex conjugate characters, label [10X30220[110X in
        [5XMOC[105X.[133X
  
  [8X[10Xpowerinfo[110X[108X
        [33X[0;6Yfield  embeddings  for  [22Xp[122X-th  symmetrizations,  [22Xp[122X  a prime integer not
        larger than the largest element order, label [10X30230[110X in [5XMOC[105X.[133X
  
  [8X[10X30900[110X[108X
        [33X[0;6Ybasic  set  of restricted ordinary irreducibles in the case of nonzero
        characteristic, all ordinary irreducibles otherwise.[133X
  
  [1X6.2-3 MOCString[101X
  
  [29X[2XMOCString[102X( [3Xmoctbl[103X[, [3Xchars[103X] ) [32X function
  
  [33X[0;0YLet [3Xmoctbl[103X be a [5XMOC[105X table record, as returned by [2XMOCTable[102X ([14X6.2-2[114X). [2XMOCString[102X
  returns a string describing the [5XMOC[105X 3 format of [3Xmoctbl[103X.[133X
  
  [33X[0;0YIf  a  second  argument  [3Xchars[103X is specified, it must be a list of [5XMOC[105X format
  characters  as  returned by [2XMOCChars[102X ([14X6.2-6[114X). In this case, these characters
  are  stored  under  label  [10X30900[110X. If the second argument is missing then the
  basic set of ordinary irreducibles is stored under this label.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmoca5:= MOCTable( CharacterTable( "A5" ) );[127X[104X
    [4X[28Xrec( 30170 := [ [  ], [ 2, 2, 1, 1 ], [ 3, 3, 1, 1 ], [ 4, 5, 1, 1 ] ][128X[104X
    [4X[28X    , [128X[104X
    [4X[28X  30900 := [ [ 1, 1, 1, 1, 0 ], [ 3, -1, 0, 0, -1 ], [128X[104X
    [4X[28X      [ 3, -1, 0, 1, 1 ], [ 4, 0, 1, -1, 0 ], [ 5, 1, -1, 0, 0 ] ], [128X[104X
    [4X[28X  GAPtbl := CharacterTable( "A5" ), centralizers := [ 60, 4, 3, 5 ], [128X[104X
    [4X[28X  cycsubgps := [ 1, 2, 3, 4, 4 ], [128X[104X
    [4X[28X  fieldbases := [128X[104X
    [4X[28X    [ CanonicalBasis( Rationals ), CanonicalBasis( Rationals ), [128X[104X
    [4X[28X      CanonicalBasis( Rationals ), [128X[104X
    [4X[28X      Basis( NF(5,[ 1, 4 ]), [ 1, E(5)+E(5)^4 ] ) ], fields := [  ], [128X[104X
    [4X[28X  galconjinfo := [ 1, 1, 2, 1, 3, 1, 4, 1, 4, 2 ], [128X[104X
    [4X[28X  identifier := "MOCTable(A5)", [128X[104X
    [4X[28X  invmap := [ [ 1, 1, 0 ], [ 1, 2, 0 ], [ 1, 3, 0 ], [128X[104X
    [4X[28X      [ 1, 4, 0, 1, 5, 0 ] ], orders := [ 1, 2, 3, 5 ], [128X[104X
    [4X[28X  powerinfo := [128X[104X
    [4X[28X    [ , [128X[104X
    [4X[28X      [ [ 1, 1, 0 ], [ 1, 1, 0 ], [ 1, 3, 0 ], [128X[104X
    [4X[28X          [ 1, 4, -1, 5, 0, -1, 5, 0 ] ], [128X[104X
    [4X[28X      [ [ 1, 1, 0 ], [ 1, 2, 0 ], [ 1, 1, 0 ], [128X[104X
    [4X[28X          [ 1, 4, -1, 5, 0, -1, 5, 0 ] ],, [128X[104X
    [4X[28X      [ [ 1, 1, 0 ], [ 1, 2, 0 ], [ 1, 3, 0 ], [ 1, 1, 0, 0 ] ] ], [128X[104X
    [4X[28X  prime := 0, repcycsub := [ 1, 2, 3, 4 ], [128X[104X
    [4X[28X  tensinfo := [128X[104X
    [4X[28X    [ [ 1 ], [ 1 ], [ 1 ], [128X[104X
    [4X[28X      [ 2, 1, 1, 1, 1, 2, 2, 0, 1, 1, 2, 1, 2, 1, -1, 2, 2, 0 ] ] )[128X[104X
    [4X[25Xgap>[125X [27Xstr:= MOCString( moca5 );;[127X[104X
    [4X[25Xgap>[125X [27Xstr{[1..68]};[127X[104X
    [4X[28X"y100y105ay110fey130t60edfy140bcdfy150bbbfcabbey160bbcbdbebecy170ccbb"[128X[104X
    [4X[25Xgap>[125X [27Xmoca5mod3:= MOCTable( CharacterTable( "A5" ) mod 3, [ 1 .. 4 ] );;[127X[104X
    [4X[25Xgap>[125X [27XMOCString( moca5mod3 ){ [ 1 .. 68 ] };[127X[104X
    [4X[28X"y100y105dy110edy130t60efy140bcfy150bbfcabbey160bbcbdbdcy170ccbbdfbby"[128X[104X
  [4X[32X[104X
  
  [1X6.2-4 ScanMOC[101X
  
  [29X[2XScanMOC[102X( [3Xlist[103X ) [32X function
  
  [33X[0;0Yreturns  a  record  containing the information encoded in the list [3Xlist[103X. The
  components  of  the  result are the labels that occur in [3Xlist[103X. If [3Xlist[103X is in
  [5XMOC[105X 2  format  (10000-format), the names of components are 30000-numbers; if
  it is in [5XMOC[105X 3 format the names of components have [10XyABC[110X-format.[133X
  
  [1X6.2-5 GAPChars[101X
  
  [29X[2XGAPChars[102X( [3Xtbl[103X, [3Xmocchars[103X ) [32X function
  
  [33X[0;0YLet [3Xtbl[103X be a character table or a [5XMOC[105X table record, and [3Xmocchars[103X be either a
  list of [5XMOC[105X format characters (as returned by [2XMOCChars[102X ([14X6.2-6[114X)) or a list of
  positive  integers  such  as  a  record  component encoding characters, in a
  record produced by [2XScanMOC[102X ([14X6.2-4[114X).[133X
  
  [33X[0;0Y[2XGAPChars[102X returns translations of [3Xmocchars[103X to [5XGAP[105X character values lists.[133X
  
  [1X6.2-6 MOCChars[101X
  
  [29X[2XMOCChars[102X( [3Xtbl[103X, [3Xgapchars[103X ) [32X function
  
  [33X[0;0YLet  [3Xtbl[103X  be a character table or a [5XMOC[105X table record, and [3Xgapchars[103X be a list
  of ([5XGAP[105X format) characters. [2XMOCChars[102X returns translations of [3Xgapchars[103X to [5XMOC[105X
  format.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xscan:= ScanMOC( str );[127X[104X
    [4X[28Xrec( y050 := [ 5, 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 1, 0, 0 ], [128X[104X
    [4X[28X  y105 := [ 0 ], y110 := [ 5, 4 ], y130 := [ 60, 4, 3, 5 ], [128X[104X
    [4X[28X  y140 := [ 1, 2, 3, 5 ], y150 := [ 1, 1, 1, 5, 2, 0, 1, 1, 4 ], [128X[104X
    [4X[28X  y160 := [ 1, 1, 2, 1, 3, 1, 4, 1, 4, 2 ], [128X[104X
    [4X[28X  y170 := [ 2, 2, 1, 1, 3, 3, 1, 1, 4, 5, 1, 1 ], [128X[104X
    [4X[28X  y210 := [ 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 0, 1, 1, 2, 1, 2, 1, -1, 2, [128X[104X
    [4X[28X      2, 0 ], y220 := [ 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 4, 0, 1, 5, 0 ],[128X[104X
    [4X[28X  y230 := [ 2, 1, 1, 0, 1, 1, 0, 1, 3, 0, 1, 4, -1, 5, 0, -1, 5, 0 ], [128X[104X
    [4X[28X  y900 := [ 1, 1, 1, 1, 0, 3, -1, 0, 0, -1, 3, -1, 0, 1, 1, 4, 0, 1, [128X[104X
    [4X[28X      -1, 0, 5, 1, -1, 0, 0 ] )[128X[104X
    [4X[25Xgap>[125X [27Xgapchars:= GAPChars( moca5, scan.y900 );[127X[104X
    [4X[28X[ [ 1, 1, 1, 1, 1 ], [ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ], [128X[104X
    [4X[28X  [ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ], [ 4, 0, 1, -1, -1 ], [128X[104X
    [4X[28X  [ 5, 1, -1, 0, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27Xmocchars:= MOCChars( moca5, gapchars );[127X[104X
    [4X[28X[ [ 1, 1, 1, 1, 0 ], [ 3, -1, 0, 0, -1 ], [ 3, -1, 0, 1, 1 ], [128X[104X
    [4X[28X  [ 4, 0, 1, -1, 0 ], [ 5, 1, -1, 0, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27XConcatenation( mocchars ) = scan.y900;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X6.3 [33X[0;0YInterface to [5XGAP[105X[101X[1X 3[133X[101X
  
  [33X[0;0YThe following functions are used to read and write character tables in [5XGAP[105X 3
  format.[133X
  
  [1X6.3-1 GAP3CharacterTableScan[101X
  
  [29X[2XGAP3CharacterTableScan[102X( [3Xstring[103X ) [32X function
  
  [33X[0;0YLet  [3Xstring[103X  be  a  string  that  contains  the output of the [5XGAP[105X 3 function
  [10XPrintCharTable[110X.  In  other  words,  [3Xstring[103X  describes  a  [5XGAP[105X  record  whose
  components   define   an   ordinary   character   table   object  in  [5XGAP[105X 3.
  [2XGAP3CharacterTableScan[102X  returns  the  corresponding  [5XGAP[105X 4  character  table
  object.[133X
  
  [33X[0;0YThe supported record components are given by the list [2XGAP3CharacterTableData[102X
  ([14X6.3-3[114X).[133X
  
  [1X6.3-2 GAP3CharacterTableString[101X
  
  [29X[2XGAP3CharacterTableString[102X( [3Xtbl[103X ) [32X function
  
  [33X[0;0YFor  an  ordinary  character  table  [3Xtbl[103X, [2XGAP3CharacterTableString[102X returns a
  string   that   when   read  into  [5XGAP[105X 3  evaluates  to  a  character  table
  corresponding  to  [3Xtbl[103X.  A  similar  format is printed by the [5XGAP[105X 3 function
  [10XPrintCharTable[110X.[133X
  
  [33X[0;0YThe supported record components are given by the list [2XGAP3CharacterTableData[102X
  ([14X6.3-3[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtbl:= CharacterTable( "Alternating", 5 );;[127X[104X
    [4X[25Xgap>[125X [27Xstr:= GAP3CharacterTableString( tbl );;[127X[104X
    [4X[25Xgap>[125X [27XPrint( str );[127X[104X
    [4X[28Xrec([128X[104X
    [4X[28Xcentralizers := [ 60, 4, 3, 5, 5 ],[128X[104X
    [4X[28Xfusions := [ rec( map := [ 1, 3, 4, 7, 7 ], name := "Sym(5)" ) ],[128X[104X
    [4X[28Xidentifier := "Alt(5)",[128X[104X
    [4X[28Xirreducibles := [[128X[104X
    [4X[28X[ 1, 1, 1, 1, 1 ],[128X[104X
    [4X[28X[ 4, 0, 1, -1, -1 ],[128X[104X
    [4X[28X[ 5, 1, -1, 0, 0 ],[128X[104X
    [4X[28X[ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ],[128X[104X
    [4X[28X[ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ][128X[104X
    [4X[28X],[128X[104X
    [4X[28Xorders := [ 1, 2, 3, 5, 5 ],[128X[104X
    [4X[28Xpowermap := [ , [ 1, 1, 3, 5, 4 ], [ 1, 2, 1, 5, 4 ], , [ 1, 2, 3, 1, \[128X[104X
    [4X[28X1 ] ],[128X[104X
    [4X[28Xsize := 60,[128X[104X
    [4X[28Xtext := "computed using generic character table for alternating groups\[128X[104X
    [4X[28X",[128X[104X
    [4X[28Xoperations := CharTableOps )[128X[104X
    [4X[25Xgap>[125X [27Xscan:= GAP3CharacterTableScan( str );[127X[104X
    [4X[28XCharacterTable( "Alt(5)" )[128X[104X
    [4X[25Xgap>[125X [27XTransformingPermutationsCharacterTables( tbl, scan );[127X[104X
    [4X[28Xrec( columns := (), group := Group([ (4,5) ]), rows := () )[128X[104X
  [4X[32X[104X
  
  [1X6.3-3 GAP3CharacterTableData[101X
  
  [29X[2XGAP3CharacterTableData[102X[32X global variable
  
  [33X[0;0YThis  is a list of pairs, the first entry being the name of a component in a
  [5XGAP[105X 3 character table and the second entry being the corresponding attribute
  name  in  [5XGAP[105X 4.  The variable is used by [2XGAP3CharacterTableScan[102X ([14X6.3-1[114X) and
  [2XGAP3CharacterTableString[102X ([14X6.3-2[114X).[133X
  
  
  [1X6.4 [33X[0;0YInterface to the Cambridge Format[133X[101X
  
  [33X[0;0YThe  following  functions deal with the so-called Cambridge format, in which
  the   source   data   of  the  character  tables  in  the  [5XAtlas[105X  of  Finite
  Groups [CCNPW85]  and in the [5XAtlas[105X of Brauer Characters [JLPW95] are stored.
  Each  such  table is stored on a file of its own. The line length is at most
  [22X78[122X,  and  each  item  of  the  table starts in a new line, behind one of the
  following prefixes.[133X
  
  [8X[10X#23[110X[108X
        [33X[0;6Ya description and the name(s) of the simple group[133X
  
  [8X[10X#7[110X[108X
        [33X[0;6Yintegers describing the column widths[133X
  
  [8X[10X#9[110X[108X
        [33X[0;6Ythe  symbols [10X;[110X and [10X@[110X, denoting columns between tables and columns that
        belong to conjugacy classes, respectively[133X
  
  [8X[10X#1[110X[108X
        [33X[0;6Ythe  symbol  [10X|[110X  in  columns  between  tables,  and  centralizer orders
        otherwise[133X
  
  [8X[10X#2[110X[108X
        [33X[0;6Ythe  symbols [10Xp[110X (in the first column only), [10Xpower[110X (in the second column
        only,  which belongs to the class of the identity element), [10X|[110X in other
        columns  between  tables,  and  descriptions  of the powers of classes
        otherwise[133X
  
  [8X[10X#3[110X[108X
        [33X[0;6Ythe  symbols [10Xp'[110X (in the first column only), [10Xpart[110X (in the second column
        only,  which belongs to the class of the identity element), [10X|[110X in other
        columns  between  tables,  and  descriptions  of  the [22Xp[122X-prime parts of
        classes otherwise[133X
  
  [8X[10X#4[110X[108X
        [33X[0;6Ythe  symbols  [10Xind[110X  and  [10Xfus[110X in columns between tables, and class names
        otherwise[133X
  
  [8X[10X#5[110X[108X
        [33X[0;6Yeither [10X|[110X or strings composed from the symbols [10X+[110X, [10X-[110X, [10Xo[110X, and integers in
        columns  where  the lines starting with [10X#4[110X contain [10Xind[110X; the symbols [10X:[110X,
        [10X.[110X,  [10X?[110X  in columns where these lines contain [10Xfus[110X; character values or [10X|[110X
        otherwise[133X
  
  [8X[10X#6[110X[108X
        [33X[0;6Ythe symbols [10X|[110X, [10Xind[110X, [10Xand[110X, and [10Xfus[110X in columns between tables; the symbol
        [10X|[110X  and  element  orders  of  preimage  classes  in downward extensions
        otherwise[133X
  
  [8X[10X#8[110X[108X
        [33X[0;6Ythe last line of the data, may contain the date of the last change[133X
  
  [8X[10X#C[110X[108X
        [33X[0;6Ycomments.[133X
  
  [1X6.4-1 CambridgeMaps[101X
  
  [29X[2XCambridgeMaps[102X( [3Xtbl[103X ) [32X function
  
  [33X[0;0YFor a character table [3Xtbl[103X, [2XCambridgeMaps[102X returns a record with the following
  components.[133X
  
  [8X[10Xname[110X[108X
        [33X[0;6Ya list of strings denoting class names,[133X
  
  [8X[10Xpower[110X[108X
        [33X[0;6Ya  list of strings, the [22Xi[122X-th entry encodes the [22Xp[122X-th powers of the [22Xi[122X-th
        class, for all prime divisors [22Xp[122X of the group order,[133X
  
  [8X[10Xprime[110X[108X
        [33X[0;6Ya  list  of  strings,  the [22Xi[122X-th entry encodes the [22Xp[122X-prime parts of the
        [22Xi[122X-th class, for all prime divisors [22Xp[122X of the group order.[133X
  
  [33X[0;0YThe  meaning  of the entries of the lists is defined in [CCNPW85, Chapter 7,
  Sections 3–5]).[133X
  
  [33X[0;0Y[2XCambridgeMaps[102X  is  used  for  example  by [2XDisplay[102X ([14XReference: Display (for a
  character  table)[114X)  in  the  case  that  the  [10Xpowermap[110X  option has the value
  [10X"ATLAS"[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCambridgeMaps( CharacterTable( "A5" ) );[127X[104X
    [4X[28Xrec( names := [ "1A", "2A", "3A", "5A", "B*" ], [128X[104X
    [4X[28X  power := [ "", "A", "A", "A", "A" ], [128X[104X
    [4X[28X  prime := [ "", "A", "A", "A", "A" ] )[128X[104X
    [4X[25Xgap>[125X [27XCambridgeMaps( CharacterTable( "A5" ) mod 2 );[127X[104X
    [4X[28Xrec( names := [ "1A", "3A", "5A", "B*" ], [128X[104X
    [4X[28X  power := [ "", "A", "A", "A" ], prime := [ "", "A", "A", "A" ] )[128X[104X
  [4X[32X[104X
  
  [1X6.4-2 StringOfCambridgeFormat[101X
  
  [29X[2XStringOfCambridgeFormat[102X( [3Xtbls[103X ) [32X function
  
  [33X[0;0Y[13X(This function is experimental.)[113X[133X
  
  [33X[0;0YLet  [3Xtbls[103X be a list of character tables, which are central extensions of the
  first  entry  in [3Xtbls[103X, and such that the factor fusion to the first entry is
  stored on all other tables in the list.[133X
  
  [33X[0;0Y[2XStringOfCambridgeFormat[102X  returns  a  string that encodes an approximation of
  the  Cambridge  format  file for the first entry in [3Xtbls[103X. Differences to the
  original  format may occur for irrational character values; the descriptions
  of  these values have been chosen deliberately for the original files, it is
  not  obvious  how to compute these descriptions from the character tables in
  question.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "A5" );;  2t:= CharacterTable( "2.A5" );;[127X[104X
    [4X[25Xgap>[125X [27XPrint( StringOfCambridgeFormat( [ t, 2t ] ) );[127X[104X
    [4X[28X#23 ? A5[128X[104X
    [4X[28X#7 4 4 4 4 4 4 [128X[104X
    [4X[28X#9 ; @ @ @ @ @ [128X[104X
    [4X[28X#1 | 60 4 3 5 5 [128X[104X
    [4X[28X#2 p power A A A A [128X[104X
    [4X[28X#3 p' part A A A A [128X[104X
    [4X[28X#4 ind 1A 2A 3A 5A B* [128X[104X
    [4X[28X#5 + 1 1 1 1 1 [128X[104X
    [4X[28X#5 + 3 -1 0 -b5 * [128X[104X
    [4X[28X#5 + 3 -1 0 * -b5 [128X[104X
    [4X[28X#5 + 4 0 1 -1 -1 [128X[104X
    [4X[28X#5 + 5 1 -1 0 0 [128X[104X
    [4X[28X#6 ind 1 4 3 5 5 [128X[104X
    [4X[28X#6 | 2 | 6 10 10 [128X[104X
    [4X[28X#5 - 2 0 -1 b5 * [128X[104X
    [4X[28X#5 - 2 0 -1 * b5 [128X[104X
    [4X[28X#5 - 4 0 1 -1 -1 [128X[104X
    [4X[28X#5 - 6 0 0 1 1 [128X[104X
    [4X[28X#8[128X[104X
  [4X[32X[104X
  
  
  [1X6.5 [33X[0;0YInterface to the [5XMAGMA[105X[101X[1X System[133X[101X
  
  [33X[0;0YThis  interface  is  intended  to  convert character tables given in [5XMAGMA[105X's
  display format into [5XGAP[105X character tables.[133X
  
  [33X[0;0YThe  function  [2XBosmaBase[102X  ([14X6.5-1[114X)  is used for the translation of irrational
  values;  this  function  may be of interest independent of the conversion of
  character tables.[133X
  
  [1X6.5-1 BosmaBase[101X
  
  [29X[2XBosmaBase[102X( [3Xn[103X ) [32X function
  
  [33X[0;0YFor  a  positive  integer  [3Xn[103X  that is not congruent to [22X2[122X modulo [22X4[122X, [2XBosmaBase[102X
  returns  the  list  of exponents [22Xi[122X for which [10XE([3Xn[103X[10X)^[110X[22Xi[122X belongs to the canonical
  basis of the [3Xn[103X-th cyclotomic field that is defined in [Bos90, Section 5].[133X
  
  [33X[0;0YAs  a  set,  this basis is defined as follows. Let [22XP[122X denote the set of prime
  divisors of [3Xn[103X and [3Xn[103X [22X= ∏_{p ∈ P} n_p[122X. Let [22Xe_l =[122X [10XE[110X[22X(l)[122X for any positive integer
  [22Xl[122X,  and  [22X{  e_{m_1}^j  }_{j  ∈  J}  ⊗  { e_{m_2}^k }_{k ∈ K} = { e_{m_1}^j ⋅
  e_{m_2}^k  }_{j  ∈  J,  k  ∈  K}[122X  for  any positive integers [22Xm_1[122X, [22Xm_2[122X. (This
  notation  is  the  same  as the one used in the description of [2XZumbroichBase[102X
  ([14XReference: ZumbroichBase[114X).)[133X
  
  [33X[0;0YThen the basis is[133X
  
  
        [33X[1;6Y[24X[33X[0;0YB_n = ⨂_{p ∈ P} B_{n_p}[133X[124X[133X
  
  
  [33X[0;0Ywhere[133X
  
  
        [33X[1;6Y[24X[33X[0;0YB_{n_p} = { e_{n_p}^k; 0 ≤ k ≤ φ(n_p)-1 };[133X[124X[133X
  
  
  [33X[0;0Yhere [22Xφ[122X denotes Euler's function, see [2XPhi[102X ([14XReference: Phi[114X).[133X
  
  [33X[0;0Y[22XB_n[122X  consists  of  roots of unity, it is an integral basis (that is, exactly
  the   integral  elements  in  [22Xℚ_n[122X  have  integral  coefficients  w.r.t. [22XB_n[122X,
  cf. [2XIsIntegralCyclotomic[102X  ([14XReference:  IsIntegralCyclotomic[114X)),  and  for any
  divisor [22Xm[122X of [3Xn[103X that is not congruent to [22X2[122X modulo [22X4[122X, [22XB_m[122X is a subset of [22XB_n[122X.[133X
  
  [33X[0;0YNote that the list [22Xl[122X, say, that is returned by [2XBosmaBase[102X is in general not a
  set.  The  ordering  of  the elements in [22Xl[122X fits to the coefficient lists for
  irrational values used by [5XMAGMA[105X's display format.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xb:= BosmaBase( 8 );[127X[104X
    [4X[28X[ 0, 1, 2, 3 ][128X[104X
    [4X[25Xgap>[125X [27Xb:= Basis( CF(8), List( b, i -> E(8)^i ) );[127X[104X
    [4X[28XBasis( CF(8), [ 1, E(8), E(4), E(8)^3 ] )[128X[104X
    [4X[25Xgap>[125X [27XCoefficients( b, Sqrt(2) );[127X[104X
    [4X[28X[ 0, 1, 0, -1 ][128X[104X
    [4X[25Xgap>[125X [27XCoefficients( b, Sqrt(-2) );[127X[104X
    [4X[28X[ 0, 1, 0, 1 ][128X[104X
    [4X[25Xgap>[125X [27Xb:= BosmaBase( 15 );[127X[104X
    [4X[28X[ 0, 5, 3, 8, 6, 11, 9, 14 ][128X[104X
    [4X[25Xgap>[125X [27Xb:= List( b, i -> E(15)^i );[127X[104X
    [4X[28X[ 1, E(3), E(5), E(15)^8, E(5)^2, E(15)^11, E(5)^3, E(15)^14 ][128X[104X
    [4X[25Xgap>[125X [27XCoefficients( Basis( CF(15), b ), EB(15) );[127X[104X
    [4X[28X[ -1, -1, 0, 0, -1, -2, -1, -2 ][128X[104X
    [4X[25Xgap>[125X [27XBosmaBase( 48 );[127X[104X
    [4X[28X[ 0, 3, 6, 9, 12, 15, 18, 21, 16, 19, 22, 25, 28, 31, 34, 37 ][128X[104X
  [4X[32X[104X
  
  [1X6.5-2 GAPTableOfMagmaFile[101X
  
  [29X[2XGAPTableOfMagmaFile[102X( [3Xfile[103X, [3Xidentifier[103X ) [32X function
  
  [33X[0;0YLet  [3Xfile[103X  be  the name of a file that contains a character table in [5XMAGMA[105X's
  display  format, and [3Xidentifier[103X be a string. [2XGAPTableOfMagmaFile[102X returns the
  corresponding [5XGAP[105X character table.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtmpdir:= DirectoryTemporary();;[127X[104X
    [4X[25Xgap>[125X [27Xfile:= Filename( tmpdir, "magmatable" );;[127X[104X
    [4X[25Xgap>[125X [27Xstr:= "\[127X[104X
    [4X[25X>[125X [27XCharacter Table of Group G\n\[127X[104X
    [4X[25X>[125X [27X--------------------------\n\[127X[104X
    [4X[25X>[125X [27X\n\[127X[104X
    [4X[25X>[125X [27X---------------------------\n\[127X[104X
    [4X[25X>[125X [27XClass |   1  2  3    4    5\n\[127X[104X
    [4X[25X>[125X [27XSize  |   1 15 20   12   12\n\[127X[104X
    [4X[25X>[125X [27XOrder |   1  2  3    5    5\n\[127X[104X
    [4X[25X>[125X [27X---------------------------\n\[127X[104X
    [4X[25X>[125X [27Xp  =  2   1  1  3    5    4\n\[127X[104X
    [4X[25X>[125X [27Xp  =  3   1  2  1    5    4\n\[127X[104X
    [4X[25X>[125X [27Xp  =  5   1  2  3    1    1\n\[127X[104X
    [4X[25X>[125X [27X---------------------------\n\[127X[104X
    [4X[25X>[125X [27XX.1   +   1  1  1    1    1\n\[127X[104X
    [4X[25X>[125X [27XX.2   +   3 -1  0   Z1 Z1#2\n\[127X[104X
    [4X[25X>[125X [27XX.3   +   3 -1  0 Z1#2   Z1\n\[127X[104X
    [4X[25X>[125X [27XX.4   +   4  0  1   -1   -1\n\[127X[104X
    [4X[25X>[125X [27XX.5   +   5  1 -1    0    0\n\[127X[104X
    [4X[25X>[125X [27X\n\[127X[104X
    [4X[25X>[125X [27XExplanation of Character Value Symbols\n\[127X[104X
    [4X[25X>[125X [27X--------------------------------------\n\[127X[104X
    [4X[25X>[125X [27X\n\[127X[104X
    [4X[25X>[125X [27X# denotes algebraic conjugation, that is,\n\[127X[104X
    [4X[25X>[125X [27X#k indicates replacing the root of unity w by w^k\n\[127X[104X
    [4X[25X>[125X [27X\n\[127X[104X
    [4X[25X>[125X [27XZ1     = (CyclotomicField(5: Sparse := true)) ! [\n\[127X[104X
    [4X[25X>[125X [27XRationalField() | 1, 0, 1, 1 ]\n\[127X[104X
    [4X[25X>[125X [27X";;[127X[104X
    [4X[25Xgap>[125X [27XFileString( file, str );;[127X[104X
    [4X[25Xgap>[125X [27Xtbl:= GAPTableOfMagmaFile( file, "MagmaA5" );;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( tbl );[127X[104X
    [4X[28XMagmaA5[128X[104X
    [4X[28X[128X[104X
    [4X[28X     2  2  2  .  .  .[128X[104X
    [4X[28X     3  1  .  1  .  .[128X[104X
    [4X[28X     5  1  .  .  1  1[128X[104X
    [4X[28X[128X[104X
    [4X[28X       1a 2a 3a 5a 5b[128X[104X
    [4X[28X    2P 1a 1a 3a 5b 5a[128X[104X
    [4X[28X    3P 1a 2a 1a 5b 5a[128X[104X
    [4X[28X    5P 1a 2a 3a 1a 1a[128X[104X
    [4X[28X[128X[104X
    [4X[28XX.1     1  1  1  1  1[128X[104X
    [4X[28XX.2     3 -1  .  A *A[128X[104X
    [4X[28XX.3     3 -1  . *A  A[128X[104X
    [4X[28XX.4     4  .  1 -1 -1[128X[104X
    [4X[28XX.5     5  1 -1  .  .[128X[104X
    [4X[28X[128X[104X
    [4X[28XA = -E(5)-E(5)^4[128X[104X
    [4X[28X  = (1-Sqrt(5))/2 = -b5[128X[104X
    [4X[25Xgap>[125X [27Xstr:= "\[127X[104X
    [4X[25X>[125X [27XCharacter Table of Group G\n\[127X[104X
    [4X[25X>[125X [27X--------------------------\n\[127X[104X
    [4X[25X>[125X [27X\n\[127X[104X
    [4X[25X>[125X [27X------------------------------\n\[127X[104X
    [4X[25X>[125X [27XClass |   1  2   3   4   5   6\n\[127X[104X
    [4X[25X>[125X [27XSize  |   1  1   1   1   1   1\n\[127X[104X
    [4X[25X>[125X [27XOrder |   1  2   3   3   6   6\n\[127X[104X
    [4X[25X>[125X [27X------------------------------\n\[127X[104X
    [4X[25X>[125X [27Xp  =  2   1  1   4   3   3   4\n\[127X[104X
    [4X[25X>[125X [27Xp  =  3   1  2   1   1   2   2\n\[127X[104X
    [4X[25X>[125X [27X------------------------------\n\[127X[104X
    [4X[25X>[125X [27XX.1   +   1  1   1   1   1   1\n\[127X[104X
    [4X[25X>[125X [27XX.2   +   1 -1   1   1  -1  -1\n\[127X[104X
    [4X[25X>[125X [27XX.3   0   1  1   J-1-J-1-J   J\n\[127X[104X
    [4X[25X>[125X [27XX.4   0   1 -1   J-1-J 1+J  -J\n\[127X[104X
    [4X[25X>[125X [27XX.5   0   1  1-1-J   J   J-1-J\n\[127X[104X
    [4X[25X>[125X [27XX.6   0   1 -1-1-J   J  -J 1+J\n\[127X[104X
    [4X[25X>[125X [27X\n\[127X[104X
    [4X[25X>[125X [27X\n\[127X[104X
    [4X[25X>[125X [27XExplanation of Character Value Symbols\n\[127X[104X
    [4X[25X>[125X [27X--------------------------------------\n\[127X[104X
    [4X[25X>[125X [27X\n\[127X[104X
    [4X[25X>[125X [27XJ = RootOfUnity(3)\n\[127X[104X
    [4X[25X>[125X [27X";;[127X[104X
    [4X[25Xgap>[125X [27XFileString( file, str );;[127X[104X
    [4X[25Xgap>[125X [27Xtbl:= GAPTableOfMagmaFile( file, "MagmaC6" );;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( tbl );[127X[104X
    [4X[28XMagmaC6[128X[104X
    [4X[28X[128X[104X
    [4X[28X     2  1  1  1  1   1   1[128X[104X
    [4X[28X     3  1  1  1  1   1   1[128X[104X
    [4X[28X[128X[104X
    [4X[28X       1a 2a 3a 3b  6a  6b[128X[104X
    [4X[28X    2P 1a 1a 3b 3a  3a  3b[128X[104X
    [4X[28X    3P 1a 2a 1a 1a  2a  2a[128X[104X
    [4X[28X[128X[104X
    [4X[28XX.1     1  1  1  1   1   1[128X[104X
    [4X[28XX.2     1 -1  1  1  -1  -1[128X[104X
    [4X[28XX.3     1  1  A /A  /A   A[128X[104X
    [4X[28XX.4     1 -1  A /A -/A  -A[128X[104X
    [4X[28XX.5     1  1 /A  A   A  /A[128X[104X
    [4X[28XX.6     1 -1 /A  A  -A -/A[128X[104X
    [4X[28X[128X[104X
    [4X[28XA = E(3)[128X[104X
    [4X[28X  = (-1+Sqrt(-3))/2 = b3[128X[104X
  [4X[32X[104X
  
