71.5 ATLAS Tables

The GAP character table library contains all character tables that are included in the ATLAS of Finite Groups (CCN85, from now on called ATLAS), and the Brauer tables contained in the ATLAS of Brauer Characters (JLPW95).

These tables have the information 

origin: ATLAS of finite groups
or 
origin: modular ATLAS of finite groups
in their InfoText value (see InfoText), they are simply called ATLAS tables further on.

For displaying ATLAS tables with the row labels used in the ATLAS, or for displaying decomposition matrices (see LaTeXStringDecompositionMatrix), see AtlasLabelsOfIrreducibles.

In addition to the information given in Chapters 6--8 of the ATLAS which tell you how to read the printed tables, there are some rules relating these to the corresponding GAP tables.

Improvements

For the GAP library not the printed ATLAS is relevant but the revised version given by the lists of Improvements to the ATLAS maintained by Simon Norton. The first list is contained in BN95, and printed in the Appendix of JLPW95; it contains the improvements that had been known until the ``ATLAS of Brauer Characters'' was published. The second list can be found in the internet, namely, an HTML version at http://www.mat.bham.ac.uk/atlas/html/atlasmods.html, and a DVI version at http://www.mat.bham.ac.uk/atlas/html/atlasmods.dvi; this list contains the improvements found since the publication of JLPW95, it is updated regularly.

Also some tables are regarded as ATLAS tables which are not printed in the ATLAS but available in ATLAS format from Cambridge, according to the lists of improvements mentioned above. Currently these are the tables related to L2(49), L2(81), L6(2), O8-(3), O8+(3), S10(2), and 2E6(2)\.3.

Power Maps

For the tables of 3\.McL, 32\.U4(3) and its covers, and 32\.U4(3)\.23 and its covers, the power maps are not uniquely determined by the information from the ATLAS but determined only up to matrix automorphisms (see MatrixAutomorphisms) of the irreducible characters. In these cases, the first possible map according to lexicographical ordering was chosen, and the automorphisms are listed in the InfoText strings of the tables.

Projective Characters and Projections

If G (or G\.a) has a nontrivial Schur multiplier then the component projectives of the GAP table object of G (or G\.a) is present, whose value is a list of records, each with the following components.

name
the Identifier value of the character table of the covering whose faithful irreducible characters are described by the record,

chars
the list of values lists of those faithful irreducibles that are printed in the ATLAS (so--called it proxy characters), and

map
a list of positions that maps each class of G to that preimage in the covering for which the column is printed in the ATLAS (a so--called it proxy class, this preimage is denoted by g0 in Chapter 7, Section 14 of the ATLAS). In a sense, a projection map is an inverse of the factor fusion from the table of the covering to the given table (see ProjectionMap). 

gap> CharacterTable( "A5" )!.projectives;
[ rec( name := "2.A5", 
      chars := [ [ 2, 0, -1, E(5)+E(5)^4, E(5)^2+E(5)^3 ], [ 2, 0, -1, 
              E(5)^2+E(5)^3, E(5)+E(5)^4 ], [ 4, 0, 1, -1, -1 ], 
          [ 6, 0, 0, 1, 1 ] ], map := [ 1, 3, 4, 6, 8 ] ) ]

Tables of Isoclinic Groups

As described in Chapter 6, Section 7 and in Chapter 7, Section 18 of the ATLAS, there exist two (in general nonisomorphic) groups of structure 2\.G\.2 for a simple group G, which are isoclinic. The table in the GAP library is the one printed in the ATLAS, the table of the isoclinic group can be constructed using CharacterTableIsoclinic (see CharacterTableIsoclinic).

Ordering of Characters and Classes

(Throughout this paragraph, G always means the involved simple group.)

1.
For G itself, the ordering of classes and characters in the GAP table coincides with the one in the ATLAS.
2.
For an automorphic extension G\.a, there are three types of characters.

If a character c of G extends to G\.a then the different extensions c0, c1, ¼, ca-1 are consecutive in the table of G\.a (see Chapter 7, Section 16 of the ATLAS).

If some characters of G fuse to give a single character of G\.a then the position of that character in the table of G\.a is given by the position of the first involved character of G.

If both extension and fusion occur occur for a character then the resulting characters are consecutive in the table of G\.a, and each replaces the first involved character of G.

3.
Similarly, there are different types of classes for an automorphic extension G\.a, as follows.

If some classes collapse then the resulting class replaces the first involved class of G.

For a > 2, any proxy class and its algebraic conjugates that are not printed in the ATLAS are consecutive in the table of G\.a; if more than two classes of G\.a have the same proxy class (the only case that actually occurs is for a = 5) then the ordering of non-printed classes is the natural one of corresponding Galois conjugacy operators *k (see Chapter 7, Section 19 in the ATLAS).

For a1, a2 dividing a such that a1 < a2, the classes of G\.a1 in G\.a precede the classes of G\.a2 not contained in G\.a1. This ordering is the same as in the ATLAS, with the only exception U3(8)\.6.

4.
For a central extension M\.G, there are two different types of characters, as follows.

Each character can be regarded as a faithful character of a factor group m\.G, where m divides M. Characters with the same kernel are consecutive as in the ATLAS, the ordering of characters with different kernels is given by the order of precedence 1, 2, 4, 3, 6, 12 for the different values of m.

If m > 2, a faithful character of m\.G that is printed in the ATLAS (a so-called it proxy character) represents two or more Galois conjugates. In each ATLAS table in GAP, a proxy character always precedes the non-printed characters with this proxy. The case m = 12 is the only one that actually occurs where more than one character for a proxy is not printed. In this case, the non-printed characters are ordered according to the corresponding Galois conjugacy operators *5, *7, *11 (in that succession).

5.
For the classes of a central extension we have the following.

The preimages of a G-class in M\.G are subsequent, the ordering is the same as that of the lifting order rows in the ATLAS (see Chapter 7, Section 7 there).

The primitive roots of unity chosen to represent the generating central element (i.e., the element in the second class of the GAP table) are E(3), E(4), E(6)^5 (= E(2) * E(3)), and E(12)^7 (= E(3) * E(4)), for m = 3, 4, 6, and 12, respectively.

6.
For tables of bicyclic extensions m\.G\.a, both the rules for automorphic and central extensions hold. Additionally we have the following three rules.

Whenever classes of the subgroup m\.G collapse (or characters of this group fuse) in m\.G\.a then the result class (or character) replaces the first involved class (or character).

Extensions of a character are subsequent, and the extensions of a proxy character precede the extensions of characters with this proxy that are not printed.

Preimages of a class of G\.a in m\.G\.a are subsequent, and the preimages of a proxy class precede the preimages of non-printed classes with this proxy.

  • AtlasLabelsOfIrreducibles( tbl[, "short"] ) F

    Let tbl be the (ordinary or Brauer) character table of a bicyclic extension of a simple group that occurs in the ATLAS of Finite Groups CCN85 or the ATLAS of Brauer Characters JLPW95. AtlasLabelsOfIrreducibles returns a list of strings, the i-th entry being a label for the i-th irreducible character of tbl.

    The labels have the following form. We state the rules only for ordinary characters, the rules for Brauer characters are obtained by replacing c by j.

    First consider only downward extensions m\.G of a simple group G. If m £ 2 then only labels of the form ci occur, which denotes the i-th ordinary character shown in the ATLAS.

    The labels of faithful ordinary characters of groups m\.G with m ³ 3 are of the form ci, ci*, or ci*k, which means the i-th character printed in the ATLAS, the unique character that is not printed and for which ci acts as proxy (see Sections 8 and 19 of Chapter 7 in the ATLAS of Finite Groups), and the image of the printed character ci under the algebraic conjugacy operator *k, respectively.

    For groups m\.G\.a with a > 1, the labels of the irreducible characters are derived from the labels of the irreducible constituents of their restrictions to m\.G, as follows.

    1.
    If the ordinary irreducible character ci of m\.G extends to m\.G\.a then the a¢ extensions are denoted by ci,0, ci,1, ¼, ci,a¢, where ci,0 is the character whose values are printed in the ATLAS.
    2.
    The label ci1 + i2 + ¼+ ia means that a different characters ci1, ci2, ¼, cia of m\.G induce to an irreducible character of m\.G\.a with this label.

    If the string "short" was entered as second argument then the label has the short form ci1+. Note that i2, i3, ¼, ia can be read off from the fusion signs in the ATLAS.

    3.
    Finally, the label ci1,j1 + i2,j2 + ¼+ ia¢,ja¢ means that the characters ci1, ci2, ¼, cia¢ of m\.G extend to a group that lies properly between m\.G and m\.G\.a, and the extensions ci1,j1, ci2,j2, ¼cia¢,ja¢ induce to an irreducible character of m\.G\.a with this label.

    Again, if the string "short" was entered as second argument then the label has a short form, namely ci,j+. 

    gap> AtlasLabelsOfIrreducibles( CharacterTable( "3.A7.2" ) );
[ "\\chi_{1,0}", "\\chi_{1,1}", "\\chi_{2,0}", "\\chi_{2,1}", "\\chi_{3+4}", 
  "\\chi_{5,0}", "\\chi_{5,1}", "\\chi_{6,0}", "\\chi_{6,1}", "\\chi_{7,0}", 
  "\\chi_{7,1}", "\\chi_{8,0}", "\\chi_{8,1}", "\\chi_{9,0}", "\\chi_{9,1}", 
  "\\chi_{17+17\\ast 2}", "\\chi_{18+18\\ast 2}", "\\chi_{19+19\\ast 2}", 
  "\\chi_{20+20\\ast 2}", "\\chi_{21+21\\ast 2}", "\\chi_{22+23\\ast 8}", 
  "\\chi_{22\\ast 8+23}" ]
gap> AtlasLabelsOfIrreducibles( CharacterTable( "3.A7.2" ), "short" );
[ "\\chi_{1,0}", "\\chi_{1,1}", "\\chi_{2,0}", "\\chi_{2,1}", "\\chi_{3+}", 
  "\\chi_{5,0}", "\\chi_{5,1}", "\\chi_{6,0}", "\\chi_{6,1}", "\\chi_{7,0}", 
  "\\chi_{7,1}", "\\chi_{8,0}", "\\chi_{8,1}", "\\chi_{9,0}", "\\chi_{9,1}", 
  "\\chi_{17+}", "\\chi_{18+}", "\\chi_{19+}", "\\chi_{20+}", "\\chi_{21+}", 
  "\\chi_{22+}", "\\chi_{23+}" ]

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    GAP 4 manual
    February 2000