67.10 Printing Character Tables

The default ViewObj (see ViewObj) method for ordinary character tables prints the string "CharacterTable", followed by the identifier (see Identifier) or, if known, the group of the character table enclosed in brackets. ViewObj for Brauer tables does the same, except that the first string is replaced by "BrauerTable", and that the characteristic is also shown.

The default PrintObj (see PrintObj) method for character tables does the same as ViewObj, except that the group is is Print-ed instead of View-ed.

The default Display (see Display) method for a character table tbl prepares the data contained in tbl for a pretty columnwise output. The number of columns printed at one time depends on the actual line length, which can be accessed and changed by the function SizeScreen (see SizeScreen).

Display shows certain characters (by default all irreducible characters) of tbl, together with the orders of the centralizers in factorized form and the available power maps (see ComputedPowerMaps). Each displayed character is given a name X.n.

The first lines of the output describe the order of the centralizer of an element of the class factorized into its prime divisors.

The next line gives the name of each class. If no class names are stored on tbl, ClassNames is called (see ClassNames).

Preceded by a name Pn, the next lines show the nth power maps of tbl in terms of the former shown class names.

Every ambiguous or unknown (see Chapter Unknowns) value of the table is displayed as a question mark ?.

Irrational character values are not printed explicitly because the lengths of their printed representation might disturb the layout. Instead of that every irrational value is indicated by a name, which is a string of at least one capital letter.

Once a name for an irrational value is found, it is used all over the printed table. Moreover the complex conjugate (see ComplexConjugate, GaloisCyc) and the star of an irrationality (see StarCyc) are represented by that very name preceded by a / and a *, respectively.

The printed character table is then followed by a legend, a list identifying the occurring symbols with their actual values. Occasionally this identification is supplemented by a quadratic representation of the irrationality together with the corresponding ATLAS--notation (see CCN85).

The optional second argument arec of Display can be used to change the default style (mentioned above) for displaying a character. arec must be a record, its relevant components are the following.

chars

an integer or a list of integers to select a sublist of the irreducible characters of tbl, or a list of characters of tbl (in this case the letter "X" is replaced by "Y"),

classes

an integer or a list of integers to select a sublist of the classes of tbl,

centralizers

suppresses the printing of the orders of the centralizers if false,

powermap

an integer or a list of integers to select a subset of the available power maps, or false to suppress the printing of power maps,

letter

a single capital letter (e. g. "P" for permutation characters) to replace "X",

indicator

true enables the printing of the second Frobenius Schur indicator, a list of integers enables the printing of the corresponding indicators (see Indicator),

StringEntry

a function that takes either a character value or a character value and the return value of StringEntryData (see below), and returns the string that is actually displayed; it is called for all character values to be displayed, and also for the displayed indicator values (see above); the default StringEntry function is CharacterTableDisplayStringEntryDefault,

StringEntryData

a unary function that is called once with argument tbl before the character values are displayed; it returns an object that is used as second argument of the function StringEntry; the default StringEntryData function is CharacterTableDisplayStringEntryDataDefault,

PrintLegend

a function that is called with the result of the StringEntryData call after the character table has been displayed; the default PrintLegend function is CharacterTableDisplayPrintLegendDefault.

gap> tbl:= CharacterTable( "A5" );;
gap> Display( tbl );
A5

     2  2  2  .  .  .
     3  1  .  1  .  .
     5  1  .  .  1  1

       1a 2a 3a 5a 5b
    2P 1a 1a 3a 5b 5a
    3P 1a 2a 1a 5b 5a
    5P 1a 2a 3a 1a 1a

X.1     1  1  1  1  1
X.2     3 -1  .  A *A
X.3     3 -1  . *A  A
X.4     4  .  1 -1 -1
X.5     5  1 -1  .  .

A = -E(5)-E(5)^4
  = (1-ER(5))/2 = -b5
gap> options:= rec( chars:= 4, classes:= [ tbl.3a .. tbl.5a ],
>                   centralizers:= false, indicator:= true,
>                   powermap:= [ 2 ] );;
gap> Indicator( tbl, 2 );;
gap> Display( tbl, options );
A5

          3a 5a
       2P 3a 5b
       2
X.4    +   1 -1
gap> SetDisplayOptions( tbl, options );  Display( tbl );
A5

          3a 5a
       2P 3a 5b
       2
X.4    +   1 -1

gap> PrintCharacterTable( CharacterTable( "Cyclic", 2 ), "tbl" );
tbl:= function()
local tbl;
tbl:=rec();
tbl.Irr:=
[ [ 1, 1 ], [ 1, -1 ] ];
tbl.NrConjugacyClasses:=
2;
tbl.Size:=
2;
tbl.OrdersClassRepresentatives:=
[ 1, 2 ];
tbl.SizesCentralizers:=
[ 2, 2 ];
tbl.UnderlyingCharacteristic:=
0;
tbl.ClassParameters:=
[ [ 1, 0 ], [ 1, 1 ] ];
tbl.CharacterParameters:=
[ [ 1, 0 ], [ 1, 1 ] ];
tbl.Identifier:=
"C2";
tbl.InfoText:=
"computed using generic character table for cyclic groups";
tbl.ComputedPowerMaps:=
[ , [ 1, 1 ] ];
ConvertToLibraryCharacterTableNC(tbl);
return tbl;
end;
tbl:= tbl();