52.6 Organization of the Table Libraries

The primary files are TBLNAME/ctadmin.tbl and TBLNAME/ctprimar.tbl. The former contains the evaluation function CharTableLibrary (see CharTable) and some utilities, the latter contains the global variable LIBLIST which encodes all information where to find library tables; the file TBLNAME/ctprimar.tbl can be constructed from the data files of the table libraries using the awk script maketbl in the etc directory of the GAP distribution.

Also the secondary files are all stored in the directory TBLNAME; they are

    clmelab.tbl   clmexsp.tbl   ctadmin.tbl   ctbalter.tbl  ctbatres.tbl
    ctbconja.tbl  ctbfisc1.tbl  ctbfisc2.tbl  ctbline1.tbl  ctbline2.tbl
    ctbline3.tbl  ctbline4.tbl  ctbline5.tbl  ctbmathi.tbl  ctbmonst.tbl
    ctborth1.tbl  ctborth2.tbl  ctborth3.tbl  ctbspora.tbl  ctbsympl.tbl
    ctbtwis1.tbl  ctbtwis2.tbl  ctbunit1.tbl  ctbunit2.tbl  ctbunit3.tbl
    ctbunit4.tbl  ctgeneri.tbl  ctoalter.tbl  ctoatres.tbl  ctocliff.tbl
    ctoconja.tbl  ctofisc1.tbl  ctofisc2.tbl  ctoholpl.tbl  ctoinert.tbl
    ctoline1.tbl  ctoline2.tbl  ctoline3.tbl  ctoline4.tbl  ctoline5.tbl
    ctoline6.tbl  ctomathi.tbl  ctomaxi1.tbl  ctomaxi2.tbl  ctomaxi3.tbl
    ctomaxi4.tbl  ctomaxi5.tbl  ctomaxi6.tbl  ctomisc1.tbl  ctomisc2.tbl
    ctomisc3.tbl  ctomisc4.tbl  ctomisc5.tbl  ctomisc6.tbl  ctomonst.tbl
    ctonews.tbl   ctoorth1.tbl  ctoorth2.tbl  ctoorth3.tbl  ctoorth4.tbl
    ctoorth5.tbl  ctospora.tbl  ctosylno.tbl  ctosympl.tbl  ctotwis1.tbl
    ctotwis2.tbl  ctounit1.tbl  ctounit2.tbl  ctounit3.tbl  ctounit4.tbl

The names start with ct for ``character table'', followed by o for ``ordinary'', b for ``Brauer'' or g for ``generic'', then an up to 5 letter description of the contents, e.g., alter for the alternating groups, and the extension .tbl.

The file ctbdescr.tbl contains (at most) the Brauer tables corresponding to the ordinary tables in ctodescr.tbl.

The format of library tables is always like this:

MOT(tblname

,
          ...
               # here the data components are stored
          ... );

Here tblname is the value of the identifier component of the table, e.g. "A5".

For the contents of the table record, there are three different ways how tables are stored:

Full tables (like that of A_5) are stored similar to the internal format (see Character Table Records). Lists of characters, however, will be abbreviated in the following way:

For each subset of characters which differ just by multiplication with a linear character or by Galois conjugacy, only one is given by its values, the others are replaced by [TENSOR,[i,j]] (which means that the character is the tensor product of the i-th and the j-th character) or [GALOIS,[i,j]] (which means that the character is the j-th Galois conjugate of the i-th character.

Brauer tables (like that of A_5 mod 2) are stored relative to the corresponding ordinary table; instead of irreducible characters the files contain decomposition matrices or Brauer trees for the blocks of nonzero defect (see Brauer Table Records), and components which can be got by restriction to p--regular classes are not stored at all.

Construction tables (like that of O_8^-(3)M7) have a component construction that is a function of one variable. This function is called by CharTable (see CharTable) when the table is constructed, i.e. not when the file containing the table is read.

The aim of this rather complicated way to store a character table is that big tables with a simple structure (e.g. direct products) can be stored in a very compact way.

Another special case where construction tables are useful is that of projective tables:

In their component irreducibles they do not contain irreducible characters but a list with information about the factor groups: Any entry is a list of length 2 that contains at position 1 the name of the table of the factor group, at the second position a list of integers representing the Galois automorphisms to get follower characters. E.g., for 12.M_{22}, the value of irreducibles is

    [["M22",[]],["2.M22",[]],
     ["3.M22",[-1,-13,-13,-1,23,23,-1,-1,-1,-1,-1]],
     ["4.M22",[-1,-1,15,15,23,23,-1,-1]],,
     ["6.M22",[-13,-13,-1,23,23,-1,-7,-7,-1,-1]],,,,,,
     ["12.M22",[[17,-17,-1],[17,-17,-1],[-55,-377,-433],[-55,-377,-433],
     [89,991,1079],[89,991,1079],[-7,7,-1]]]]

Using this and the projectives component of the table of the smallest nontrivial factor group, CharTable CharTable constructs the irreducible characters. The table head, however, need not be constructed.

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