2 The GAP Character Table Library

If the only argument is a string tblname and if this is an admissible name (see below) of a library character table then CharacterTableFromLibrary returns this library table, otherwise fail.

If CharacterTableFromLibrary is called with more than one argument then the first must be a string series specifying a series of groups which is implemented via a generic character table, for example "Symmetric" for symmetric groups; the remaining arguments specialise then the desired member of the series (see Generic Character Tables for a list of available generic tables). If no generic table with name series is available or if the parameters are not admissible then CharacterTableFromLibrary returns fail.

A call of CharacterTableFromLibrary may cause to read some library files and to construct the table object from the data stored in these files, so fetching a library table may take more time than on expects.

CharacterTableFromLibrary is called by CharacterTable if the first argument is a string, so one may also call CharacterTable.

Admissible names for the ordinary character table t of the group G are

-

an ATLAS like name if t is an ATLAS table (see ATLAS Tables), for example "M22" for the table of the Mathieu group M₂₂, "L2(13).2" for L₂(13):2, and "12_1.U4(3).2_1" for 12₁.U₄(3).2₁,

(The difference to the name printed in the ATLAS is that subscripts and superscripts are omitted except if they are used to qualify integer values, and double dots are replaced by a single dot.)

-

the names that were admissible for tables of G in CAS if the CAS table library contained a table of G, for example sl42 for the table of the alternating group A₈,

(But note that the GAP table may be different from that in CAS, see CAS Tables.)

-

some ``relative'' names, as follows.

If G is the n--th maximal subgroup (in decreasing group order) of a group whose library table s is available in GAP and stores the Maxes value (see Maxes), and if name is an admissible name for s then nameMn is admissible for t. For example, the name "J3M2" can be used to access the second maximal subgroup of the sporadic simple Janko group J₃ which has the admissible name J3.

If G is a nontrivial Sylow p normalizer in a sporadic simple group with admissible name name, --where nontrivial means that G is not isomorphic to a subgroup of p:(p-1)-- then nameNp is an admissible name of t. For example, the name "J4N11" can be used to access the table of the Sylow 11 normalizer in the sporadic simple Janko group J₄.

In a few cases, the table of the Sylow p subgroup of G is accessible via the name nameSylp where name is an admissible name of the table of G. For example, "A11Syl2" is an admissible name for the table of the Sylow 2 subgroup of the alternating group A₁₁.

In a few cases, the table of an element centralizer in G is accessible via the name nameCcl where name is an admissible name of the table of G. For example, "M11C2" is an admissible name for the table of an involution centralizer in the Mathieu group M₁₁.

The recommended way to access Brauer tables from the library is via the mod operator from the ordinary table and the desired characteristic (see BrauerTable and Operators for Character Tables in the GAP Reference Manual), so it is not necessary to define admissible names of Brauer tables.

A generic character table (see Generic Character Tables) is accessible only by the name given by its Identifier value (see Identifier!for character tables in the GAP Reference Manual).

Case is not significant for character table names. For example, both "suzm3" and "SuzM3" are admissible names for the third maximal subgroup of the sporadic simple Suzuki group.

gap> s5:= CharacterTable( "A5.2" );
CharacterTable( "A5.2" )
gap> sym5:= CharacterTable( "Symmetric", 5 );
CharacterTable( "Sym(5)" )
gap> TransformingPermutationsCharacterTables( s5, sym5 );
rec( columns := (2,3,4,7,5), rows := (1,7,3,4,6,5,2), group := Group(()) )

gap> CharacterTable( "J5" );
fail
gap> CharacterTable( "A5" ) mod 2;
BrauerTable( "A5", 2 )

is a list of identifiers of the tables of all maximal subgroups of tbl. This is meaningful usually only for library tables, and there is no default method to compute the value.

If the Maxes value of tbl is stored then it lists exactly one representative for each conjugacy class of maximal subgroups of the group of tbl; the tables of these maximal subgroups are then available in the GAP table library, and the fusions to tbl are stored on these tables.

gap> tbl:= CharacterTable( "M11" );;
gap> HasMaxes( tbl );
true
gap> maxes:= Maxes( tbl );
[ "A6.2_3", "L2(11)", "3^2:Q8.2", "A5.2", "2.S4" ]
gap> CharacterTable( maxes[1] );
CharacterTable( "A6.2_3" )

Similar to group libraries (see Chapter Group Libraries in the GAP Reference Manual), the GAP character table library can be used to search for ordinary character tables with prescribed properties.

A specific library table can be selected by an admissible name (see CharacterTableFromLibrary).

The selection function for character tables with certain abstract properties is AllCharacterTableNames. Contrary to the situation in the case of group libraries, the selection function returns a list not of library character tables but of their names; using CharacterTable one can then access the tables themselves.

AllCharacterTableNames takes an arbitrary even number of arguments. The argument at each odd position must be a function, and the argument at the subsequent even position must be a value that this function must return when called for the character table in question, in order to have the name of the table included in the selection, or a list of such values. For example,

gap> names:= AllCharacterTableNames();;

gap> simpnames:= AllCharacterTableNames( IsSimple, true );;
gap> AllCharacterTableNames( IsSimple, true, Size, [ 1 .. 100 ] );
[ "A5" ]

For the sake of efficiency, the arguments IsSimple and IsSporadicSimple followed by true are handled in a special way, GAP need not read all files of the table library in these cases in order to find the desired names.

If the function OfThose is an argument at an odd position then the following argument func must be a function that takes a character table and returns a name of a character table or a list of names; this is interpreted as replacement of the names computed up to this position by the union of names returned by func. For example, func may be Maxes (see Maxes) or NamesOfFusionSources (see NamesOfFusionSources in the GAP Reference Manual).

gap> maxesnames:= AllCharacterTableNames( IsSporadicSimple, true,
>                                         HasMaxes, true,
>                                         OfThose, Maxes );;

2.3 Generic Character Tables

Generic character tables provide a means for writing down the character tables of all groups in a (usually infinite) series of similar groups, e.g., cyclic groups, or symmetric groups, or the general linear groups GL(2,q) where q ranges over certain prime powers.

Let { G_q || q Î I } be such a series, where I is an index set. The character table of one fixed member G_q could be computed using a function that takes q as only argument and constructs the table of G_q. It is, however, often desirable to compute not only the whole table but to access just one specific character, or to compute just one character value, without computing the whole character table.

For example, both the conjugacy classes and the irreducible characters of the symmetric group S_n are in bijection with the partitions of n. Thus for given n it makes sense to ask for the character corresponding to a particular partition, or just for its character value at a partition.

A generic character table in GAP allows one such local evaluations. In this sense, GAP can deal also with character tables that are too big to be computed and stored as a whole.

Currently the only operations for generic tables supported by GAP are the specialisation of the parameter q in order to compute the whole character table of G_q, and local evaluation (see ClassParameters for an example). It is not possible to compute, e.g., generic scalar products.

Currently, generic tables of the following groups --in alphabetical order-- are available in GAP. (A list of the names of generic tables known to GAP is LIBTABLE.GENERIC.firstnames.) We list the function calls needed to get a specialized table, the generic table itself can be accessed by calling CharacterTable with the first argument only; for example, CharacterTable( "Cyclic" ) yields the generic table of cyclic groups.

CharacterTable( "Alternating", n ), the table of the alternating group on n letters,

CharacterTable( "Cyclic", n ), the table of the cyclic group of order n,

CharacterTable( "Dihedral", 2n ), the table of the dihedral group of order 2n,

CharacterTable( "GL", 2, q ), the table of the general linear group GL(2,q), for a prime power q,

CharacterTable( "GU", 3, q ), the table of the general unitary group GU(3,q), for a prime power q,

CharacterTable( "P:Q", [ p, q ] ), the table of the Frobenius extension of the cyclic group of prime order p by a cyclic group of order q where q divides p-1,

CharacterTable( "PSL", 2, q ), the table of the projective special linear group PSL(2,q), for a prime power q,

CharacterTable( "SL", 2, q ), the table of the special linear group SL(2,q), for a prime power q,

CharacterTable( "SU", 3, q ), the table of the special unitary group SU(3,q), for a prime power q,

CharacterTable( "Suzuki", q ), the table of the Suzuki group Sz(q) = ²B₂(q), for q an odd power of 2,

CharacterTable( "Symmetric", n ), the table of the symmetric group on n letters,

CharacterTable( "WeylB", n ), the table of the Weyl group of type B_n,

CharacterTable( "WeylD", n ), the table of the Weyl group of type D_n.

CharacterTable( "Quaternionic", 4n ), the table of the quaternionic (dicyclic) group of order 4n,

CharacterTableWreathSymmetric( tbl, n ), the character table of the wreath product of the group whose table is tbl with the symmetric group on n letters (see CharacterTableWreathSymmetric in the GAP Reference Manual).

For a record generic_table representing a generic character table, and a parameter value q, CharacterTableSpecialized returns a character table object computed by evaluating generic_table at q.

gap> c5:= CharacterTableSpecialized( CharacterTable( "Cyclic" ), 5 );
CharacterTable( "C5" )
gap> Display( c5 );
C5

     5  1  1  1  1  1

       1a 5a 5b 5c 5d
    5P 1a 1a 1a 1a 1a

X.1     1  1  1  1  1
X.2     1  A  B /B /A
X.3     1  B /A  A /B
X.4     1 /B  A /A  B
X.5     1 /A /B  B  A

A = E(5)
B = E(5)^2

While the numbers of conjugacy classes for the members of a series of groups are usually not bounded, there is always a fixed finite number of types (equivalence classes) of conjugacy classes; very often the equivalence relation is isomorphism of the centralizers of the representatives.

For each type t of classes and a fixed q Î I, a parametrisation of the classes in t is a function that assigns to each conjugacy class of G_q in t a parameter by which it is uniquely determined. Thus the classes are indexed by pairs [t,p_t] consisting of a type t and a parameter p_t for that type.

For any generic table, there has to be a fixed number of types of irreducible characters of G_q, too. Like the classes, the characters of each type are parametrised.

In GAP, the parametrisations of classes and characters for tables computed from generic tables is stored using the attributes ClassParameters and CharacterParameters.

are lists containing a parameter for each conjugacy class or irreducible character, respectively, of the character table tbl.

It depends on tbl what these parameters are, so there is no default to compute class and character parameters.

For example, the classes of symmetric groups can be parametrized by partitions, corresponding to the cycle structures of permutations. Character tables constructed from generic character tables (see Generic Character Tables) usually have class and character parameters stored.

If tbl is a p-modular Brauer table such that class parameters are stored in the underlying ordinary table (see OrdinaryCharacterTable in the GAP Reference Manual) of tbl then ClassParameters returns the sublist of class parameters of the ordinary table, for p-regular classes.

gap> HasClassParameters( c5 );  HasCharacterParameters( c5 );
true
true
gap> ClassParameters( c5 );  CharacterParameters( c5 );
[ [ 1, 0 ], [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 4 ] ]
[ [ 1, 0 ], [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 4 ] ]
gap> ClassParameters( CharacterTable( "Symmetric", 3 ) );
[ [ 1, [ 1, 1, 1 ] ], [ 1, [ 2, 1 ] ], [ 1, [ 3 ] ] ]

gap> CharacterTable( "Cyclic" ).irreducibles[1][1]( 5, 2, 3 );
E(5)
gap> tbl:= CharacterTable( "Symmetric" );;
gap> tbl.irreducibles[1][1]( 5, [ 3, 2 ], [ 2, 2, 1 ] );
1
gap> tbl.orders[1]( 5, [ 2, 1, 1, 1 ] );
2

Any generic table in GAP is represented by a record. The following components are supported for generic character table records.

centralizers

list of functions, one for each class type t, with arguments q and p_t, returning the centralizer order of the class [t,p_t],

charparam

list of functions, one for each character type t, with argument q, returning the list of character parameters of type t,

classparam

list of functions, one for each class type t, with argument q, returning the list of class parameters of type t,

classtext

list of functions, one for each class type t, with arguments q and p_t, returning a representative of the class with parameter [t,p_t],

domain

function of q returning true if q is a valid parameter, and false otherwise,

identifier

identifier string of the generic table,

irreducibles

list of list of functions, in row i and column j the function of three arguments, namely q and the parameters p_t and p_s of the class type t and the character type s,

isGenericTable

always true

libinfo

record with components firstname (Identifier value of the table) and othernames (list of other admissible names)

matrix

function of q returning the matrix of irreducibles of G_q,

orders

list of functions, one for each class type t, with arguments q and p_t, returning the representative order of elements of type t and parameter p_t,

powermap

list of functions, one for each class type t, each with three arguments q, p_t, and k, returning the pair [s,p_s] of type and parameter for the k-th power of the class with parameter [t,p_t],

size

function of q returning the order of G_q,

specializedname

function of q returning the Identifier value of the table of G_q,

text

string informing about the generic table

In the specialized table, the ClassParameters and CharacterParameters values are the lists of parameters [t,p_t] of classes and characters, respectively.

If the matrix component is present then its value implements a method to compute the complete table of small members G_q more efficiently than via local evaluation; this method will be called when the generic table is used to compute the whole character table for a given q (see CharacterTableSpecialized).

2.4 Examples of Generic Character Tables

1. The generic table of cyclic groups.

For the cyclic group C_q = áx ñ of order q, there is one type of classes. The class parameters are integers k Î { 0, ¼, q-1 }, the class with parameter k consists of the group element x^k. Group order and centralizer orders are the identity function q ® q, independent of the parameter k. The representative order function maps the parameter pair [q,k] to [(q)/(gcd(q,k))], which is the order of x^k in C_q; the p-th power map is the function mapping the triple (q,k,p) to the parameter [1,(kp mod q)].

There is one type of characters, with parameters l Î { 0, ¼, q-1 }; for e_q a primitive complex q-th root of unity, the character values are c_l(x^k) = e_q^kl.

The library file contains the following generic table.

rec(
identifier := "Cyclic",
specializedname := ( q -> Concatenation( "C", String(q) ) ),
size := ( n -> n ),
text := "generic character table for cyclic groups",
centralizers := [ function( n, k ) return n; end ],
classparam := [ ( n -> [ 0 .. n-1 ] ) ],
charparam := [ ( n -> [ 0 .. n-1 ] ) ],
powermap := [ function( n, k, pow ) return [ 1, k*pow mod n ]; end ],
orders := [ function( n, k ) return n / Gcd( n, k ); end ],
irreducibles := [ [ function( n, k, l ) return E(n)^(k*l); end ] ],
domain := IsPosInt,
libinfo := rec( firstname:= "Cyclic", othernames:= [] ),
isGenericTable := true )

2. The generic table of the general linear group GL(2,q).

We have four types t₁, t₂, t₃, t₄ of classes, according to the rational canonical form of the elements. t₁ describes scalar matrices, t₂ nonscalar diagonal matrices, t₃ companion matrices of (X-r)² for elements r Î F_q^*, and t₄ companion matrices of irreducible polynomials of degree 2 over F_q.

The sets of class parameters of the types are in bijection with F_q^* for t₁ and t₃, with the set {{r,t}; r, t Î F_q^*, r ¹ t} for t₂, and with the set {{e,e^q}; e Î F_q²\F_q} for t₄.

The centralizer order functions are q ® (q²-1)(q²-q) for type t₁, q ® (q-1)² for type t₂, q ® q(q-1) for type t₃, and q ® q²-1 for type t₄.

The representative order function of t₁ maps (q,r) to the order of r in F_q, that of t₂ maps (q,{r,t}) to the least common multiple of the orders of r and t.

The file contains something similar to the following table.

rec(
identifier := "GL2",
specializedname := ( q -> Concatenation( "GL(2,", String(q), ")" ) ),
size := ( q -> (q^2-1)*(q^2-q) ),
text := "generic character table of GL(2,q), see Robert Steinberg: ...",
centralizers := [ function( q, k ) return (q^2-1) * (q^2-q); end,
                  ..., ..., ... ],
classparam := [ ( q -> [ 0 .. q-2 ] ), ..., ..., ... ],
charparam := [ ( q -> [ 0 .. q-2 ] ), ..., ..., ... ],
powermap := [ function( q, k, pow ) return [ 1, (k*pow) mod (q-1) ]; end,
              ..., ..., ... ],
orders:= [ function( q, k ) return (q-1)/Gcd( q-1, k ); end,
           ..., ..., ... ],
irreducibles := [ [ function( q, k, l ) return E(q-1)^(2*k*l); end,
                    ..., ..., ... ],
                  [ ..., ..., ..., ... ],
                  [ ..., ..., ..., ... ],
                  [ ..., ..., ..., ... ] ],
classtext := [ ..., ..., ..., ... ],
domain := ( q -> IsInt(q) and q > 1 and Length( Set( FactorsInt(q) ) ) = 1 ),
isGenericTable := true )

2.5 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

origin: modular ATLAS of finite groups

For displaying ATLAS tables with the row labels used in the ATLAS, or for displaying decomposition matrices, see LaTeXStringDecompositionMatrix in the GAP Reference Manual and 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 in the GAP Reference Manual) 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 projective irreducibles that are printed in the ATLAS (so--called 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 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 in the GAP Reference Manual).

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 in the GAP Reference Manual).

Ordering of Characters and Classes

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

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, c^a-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 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 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 in m.G.a then the resulting class replaces the first involved class.

Whenever characters of the subgroup m.G collapse fuse in m.G.a then the result character replaces the first involved 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.

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 c_i 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 c_i, c_i^*, or c_i^*k, which means the i-th character printed in the ATLAS, the unique character that is not printed and for which c_i acts as proxy (see Sections 8 and 19 of Chapter 7 in the ATLAS of Finite Groups), and the image of the printed character c_i 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 c_i of m.G extends to m.G.a then the a^¢ extensions are denoted by c_i,0, c_i,1, ¼, c_i,a^¢, where c_i,0 is the character whose values are printed in the ATLAS.

2.

The label c_{i1 + i2 + ¼+ ia} means that a different characters c_i1, c_i2, ¼, c_ia of m.G induce to an irreducible character of m.G.a with this label.

If the string "short" was entered as the second argument then the label has the short form c_i1+. Note that i₂, i₃, ¼, i_a can be read off from the fusion signs in the ATLAS.

3.

Finally, the label c_{i1,j1 + i2,j2 + ¼+ ia^¢,ja^¢} means that the characters c_i1, c_i2, ¼, c_ia^¢ of m.G extend to a group that lies properly between m.G and m.G.a, and the extensions c_i1,j1, c_i2,j2, ¼c_ia^¢,ja^¢ induce to an irreducible character of m.G.a with this label.

Again, if the string "short" was entered as the second argument then the label has a short form, namely c_i,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+}" ]

2.6 Examples of the ATLAS Format for GAP Tables

We give three little examples for the conventions stated in ATLAS Tables, listing both the ATLAS format and the table displayed by GAP.

First, let G be the trivial group. We consider the cyclic group C₆ of order 6. It can be viewed in several ways, namely

1.

as a downward extension of the factor group C₂ which contains G as a subgroup, or equivalently, as an upward extension of the subgroup C₃ which has a factor group isomorphic to G,

2.

as a downward extension of the factor group C₃ which contains G as a subgroup, or equivalently, as an upward extension of the subgroup C₂ which has a factor group isomorphic to G,

3.

as a downward extension of the factor groups C₃ and C₂ which have G as a factor group, or

4.

as an upward extension of the subgroups C₃ or C₂ which both contain a subgroup isomorphic to G.

Situation 1. is shown here.

 -------   -------         ;   @   ;   ;   @      2   1   1   1   1   1   1
|       | |       |            1           1      3   1   1   1   1   1   1
|   G   | |  G.2  |      p power           A
|       | |       |      p' part           A         1a  3a  3b  2a  6a  6b
 -------   -------       ind  1A fus ind  2A     2P  1a  3b  3a  1a  3b  3a
 -------   -------                               3P  1a  1a  1a  2a  2a  2a
|       | |       |  chi1  +   1   :  ++   1
|  3.G  | | 3.G.2 |                            X.1    1   1   1   1   1   1
|       | |       |      ind   1 fus ind   2   X.2    1   1   1  -1  -1  -1
 -------   -------             3           6   X.3    1   A  /A   1   A  /A
                               3           6   X.4    1   A  /A  -1  -A -/A
                                               X.5    1  /A   A   1  /A   A
                     chi2 o2   1   : oo2   1   X.6    1  /A   A  -1 -/A  -A

                                               A = E(3)
                                                 = (-1+ER(-3))/2 = b3

Situation 2. is shown here.

 -------   -------         ;   @   ;   ;   @      2   1   1   1   1   1   1
|       | |       |            1           1      3   1   1   1   1   1   1
|   G   | |  G.3  |      p power           A
|       | |       |      p' part           A         1a  2a  3a  6a  3b  6b
 -------   -------       ind  1A fus ind  3A     2P  1a  1a  3b  3b  3a  3a
 -------   -------                               3P  1a  2a  1a  2a  1a  2a
|       | |       |  chi1  +   1   : +oo   1
|  2.G  | | 2.G.3 |                            X.1    1   1   1   1   1   1
|       | |       |      ind   1 fus ind   3   X.2    1   1   A   A  /A  /A
 -------   -------             2           6   X.3    1   1  /A  /A   A   A
                                               X.4    1  -1   1  -1   1  -1
                     chi2  +   1   : +oo   1   X.5    1  -1   A  -A  /A -/A
                                               X.6    1  -1  /A -/A   A  -A

                                               A = E(3)
                                                 = (-1+ER(-3))/2 = b3

Situation 3. is shown here.

 -------             ;   @        2   1   1   1   1   1   1
|       |                1        3   1   1   1   1   1   1
|   G   |          p power
|       |          p' part           1a  6a  3a  2a  3b  6b
 -------           ind  1A       2P  1a  3a  3b  1a  3a  3b
 -------                         3P  1a  2a  1a  2a  1a  2a
|       |      chi1  +   1
|  2.G  |                      X.1    1   1   1   1   1   1
|       |          ind   1     X.2    1  -1   1  -1   1  -1
 -------                 2     X.3    1   A  /A   1   A  /A
 -------                       X.4    1  /A   A   1  /A   A
|       |      chi2  +   1     X.5    1  -A  /A  -1   A -/A
|  3.G  |                      X.6    1 -/A   A  -1  /A  -A
|       |          ind   1
 -------                 3     A = E(3)
 -------                 3       = (-1+ER(-3))/2 = b3
|       |
|  6.G  |      chi3 o2   1
|       |
 -------           ind   1
                         6
                         3
                         2
                         3
                         6

               chi4 o2   1

Finally, situation 4. is shown here.

 -------   -------   -------   -------
|       | |       | |       | |       |
|   G   | |  G.2  | |  G.3  | |  G.6  |
|       | |       | |       | |       |
 -------   -------   -------   -------

      ;   @   ;   ;   @   ;   ;   @   ;     ;   @

          1           1           1             1
    p power           A           A            AA
    p' part           A           A            AA
    ind  1A fus ind  2A fus ind  3A fus   ind  6A

chi1  +   1   :  ++   1   : +oo   1   :+oo+oo   1


    2   1   1   1   1   1   1
    3   1   1   1   1   1   1

       1a  2a  3a  3b  6a  6b
   2P  1a  1a  3b  3a  3b  3a
   3P  1a  2a  1a  1a  2a  2a
 X.1    1   1   1   1   1   1
 X.2    1  -1   A  /A  -A -/A
 X.3    1   1  /A   A  /A   A
 X.4    1  -1   1   1  -1  -1
 X.5    1   1   A  /A   A  /A
 X.6    1  -1  /A   A -/A  -A

 A = E(3)
   = (-1+ER(-3))/2 = b3

The second example explains the fusion case; again, G is the trivial group.

 -------   -------        ;   @   ;   ;  @      3.G.2
|       | |       |           1          1
|   G   | |  G.2  |     p power          A         2   1   .   1
|       | |       |     p' part          A         3   1   1   .
 -------   -------      ind  1A fus ind 2A
 -------   -------                                    1a 3a 2a
|       | |       |   X1  +   1   :  ++  1        2P  1a 3a 1a
|  2.G  | | 2.G.2 |                               3P  1a 1a 2a
|       | |       |     ind   1 fus ind  2
 -------   -------            2          2      X.1    1  1  1
 -------   -------                              X.2    1  1 -1
|       | |           X2  +   1   :  ++  1      X.3    2 -1  .
|  3.G  | | 3.G.2
|       | |             ind   1 fus ind  2
 -------                      3                 6.G.2
 -------   -------            3
|       | |                                        2   2  1  1  2  2  2
|  6.G  | | 6.G.2     X3 o2   1   *   +            3   1  1  1  1  .  .
|       | |
 -------                ind   1 fus ind  2            1a 6a 3a 2a 2b 2c
                              6          2        2P  1a 3a 3a 1a 1a 1a
                              3                   3P  1a 2a 1a 2a 2b 2c
                              2
                              3                 Y.1    1  1  1  1  1  1
                              6                 Y.2    1  1  1  1 -1 -1
                                                Y.3    1 -1  1 -1  1 -1
                      X4 o2   1   *   +         Y.4    1 -1  1 -1 -1  1
                                                Y.5    2 -1 -1  2  .  .
                                                Y.6    2  1 -1 -2  .  .

In 3.G.2, the characters X.1, X.2 extend c₁; c₃ and its non-printed partner fuse to give X.3, and the two preimages of 1A of order 3 collapse.

In 6.G.2, Y.1--Y.4 are extensions of c₁, c₂, so these characters are the inflated characters from 2.G.2 (with respect to the factor fusion [ 1, 2, 1, 2, 3, 4 ]). Y.5 is inflated from 3.G.2 (with respect to the factor fusion [ 1, 2, 2, 1, 3, 3 ]), and Y.6 is the result of the fusion of c₄ and its non-printed partner.

For the last example, let G be the elementary abelian group 2² of order 4. Consider the following tables.

 -------   -------            ;   @   @   @   @   ;   ;   @
|       | |       |               4   4   4   4           1
|   G   | |  G.3  |         p power   A   A   A           A
|       | |       |         p' part   A   A   A           A
 -------   -------          ind  1A  2A  2B  2C fus ind  3A
 -------   -------
|       | |       |     chi1  +   1   1   1   1   : +oo   1
|  2.G  | | 2.G.3 |     chi2  +   1   1  -1  -1   .   +   0
|       | |       |     chi3  +   1  -1   1  -1   |
 -------   -------      chi4  +   1  -1  -1   1   |

                            ind   1   4   4   4 fus ind   3
                                  2                       6

                        chi5  -   2   0   0   0   : -oo   1

  G.3

     2   2   2   .   .
     3   1   .   1   1

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

  X.1    1   1   1   1
  X.2    1   1   A  /A
  X.3    1   1  /A   A
  X.4    3  -1   .   .

  A = E(3)
    = (-1+ER(-3))/2 = b3

  2.G                          2.G.3

     2   3   3   2   2   2        2   3   3   2   1   1   1   1
                                  3   1   1   .   1   1   1   1
        1a  2a  4a  4b  4c
    2P  1a  1a  2a  1a  1a           1a  2a  4a  3a  6a  3b  6b
    3P  1a  2a  4a  4b  4c       2P  1a  1a  2a  3b  3b  3a  3a
                                 3P  1a  2a  4a  1a  2a  1a  2a
  X.1    1   1   1   1   1
  X.2    1   1   1  -1  -1     X.1    1   1   1   1   1   1   1
  X.3    1   1  -1   1  -1     X.2    1   1   1   A   A  /A  /A
  X.4    1   1  -1  -1   1     X.3    1   1   1  /A  /A   A   A
  X.5    2  -2   .   .   .     X.4    3   3  -1   .   .   .   .
                               X.5    2  -2   .   1   1   1   1
                               X.6    2  -2   .   A  -A  /A -/A
                               X.7    2  -2   .  /A -/A   A  -A

                               A = E(3)
                                 = (-1+ER(-3))/2 = b3

(The beautiful LaTeX pictures that were part of the GAP 3 manual will be reintroduced as soon as the decision to use TeX for the manual will be revised.)

2.7 CAS Tables

All character tables of the CAS table library (see NPP84) are available in GAP. This sublibrary has been completely revised before it was included in GAP, for example, errors have been corrected and power maps have been completed.

Any CAS table is accessible by each of its CAS names, that is, the table name or the filename used in CAS.

gap> tbl:= CharacterTable( "m10" );
CharacterTable( "A6.2_3" )

Let tbl be an ordinary character table tbl in the GAP library that was contained already in the CAS table library. When one fetches tbl from the library, one does in general not get the original CAS table. Namely, in many cases (mostly ATLAS tables, see ATLAS Tables) the identifier of the table (see Identifier!for character tables in the GAP Reference Manual) as well as the ordering of classes and characters are different for the CAS table and its GAP version.

Note that in several cases, the CAS library contains different tables of the same group, in particular these tables may have different names and orderings of classes and characters.

The CASInfo value of tbl, if stored, is a list of records, each describing the relation between tbl and a character table in the CAS library. The records have the components

name

the name of the CAS table,

permchars and permclasses

permutations of the Irr values and the classes of tbl, respectively, that must be applied in order to get the orderings in the original CAS table, and

text

the text that was stored on the CAS table (which may contain incorrect statements).

gap> HasCASInfo( tbl );
true
gap> CASInfo( tbl );
[ rec( name := "m10", permchars := (3,5)(4,8,7,6), permclasses := (), 
      text := "names:     m10\norder:     2^4.3^2.5 = 720\nnumber of classes: \
8\nsource:    cambridge atlas\ncomments:  point stabilizer of mathieu-group m1\
1\ntest:      orth, min, sym[3]\n" ) ]

The class fusions stored on tables from the CAS library have been computed anew; the text component of such a fusion record tells if the fusion map is equal to that in the CAS library --of course modulo the permutation of classes between the table in CAS and its GAP version.

gap> First( ComputedClassFusions( tbl ), x -> x.name = "M11" );
rec( name := "M11", map := [ 1, 2, 3, 4, 5, 4, 7, 8 ], 
  text := "fusion is unique up to table automorphisms,\nthe representative is \
equal to the fusion map on the CAS table" )

2.8 Organization of the Character Table Library

The data files of the GAP character table library reside in the data directory of the ctbllib package.

The filenames start with ct (for ``character table''), followed by either o (for ``ordinary''), b (for ``Brauer''), or g (for ``generic''), then a description of the contents (up to 5 characters, e.g., alter for the tables of alternating and related groups), and the suffix .tbl.

The file ctbdescr.tbl contains the known Brauer tables corresponding to the ordinary tables in the file ctodescr.tbl.

Each data file of the table library is supposed to consist of

1.

comment lines, starting with in the first column,

2.

assignments to ALN (short for ``add library name'', see NotifyNameOfCharacterTable) and to a component of Revision, at the beginning of the file, for example in the file with name ctoalter.tbl a value is assigned to Revision.ctoalter_tbl,

3.

assignments to ALN and to a component of LIBTABLE.LOADSTATUS, at the end of the file, and

4.

function calls of the form SET_TABLEFILENAME( filename );, MBT( name, data ); (``make Brauer table''), MOT( name, data ); (``make ordinary table''), ALF( from, to, map );, ALF( from, to, map, textlines ); (``add library fusion''), ALN( name, listofnames );, and ARC( name, component, compdata ); (``add record component'').

Here filename must be a string corresponding to the filename but without suffix, for example "ctoalter" if the file has the name ctoalter.tbl; name must be the identifier value of the ordinary character table corresponding to the table to which the command refers; data must be a comma separated sequence of GAP objects; from and to must be identifier values of ordinary character tables, map a list of positive itegers, textlines and listofnames lists list of strings, component a string, and compdata any GAP object.

MOT, ALF, ALN, and ARC occur only in files containing ordinary character tables, and MBT occurs only in files containing Brauer tables.

ACM, Concatenation, ConstructClifford, ConstructDirectProduct, ConstructFactor, ConstructGS3, ConstructIsoclinic, ConstructMixed, ConstructPermuted, ConstructProj, ConstructSubdirect, ConstructV4G, E, EvalChars, GALOIS, Length, NotifyCharTableName, ShallowCopy, TENSOR, and TransposedMat.

The awk script maketbl in the etc directory of the ctbllib package expects the file format described above, and to some extent this format is checked by this script.

The function calls may be continued over several lines of a file. The ; is assumed to be the last character in its line if and only if it terminates a function call.

Names of character tables are strings (see Chapter Strings and Characters in the GAP Reference Manual), i.e., they are enclosed in double quotes; strings in table library files must not be split over several lines, because otherwise the awk script may get confused. Additionally, no character table name is allowed to contain double quotes.

The knowledge of GAP about the ordinary tables in the table library is contained in the file ctprimar.tbl (the ``primary file'' of the table library). This file is automatically produced from the library files by the script maketbl in the etc directory of the ctbllib package. The information is stored in the global variable LIBLIST, which is a record with the following components.

firstnames

the list of Identifier (see Identifier!for character tables in the GAP Reference Manual) values of the ordinary tables,

files

the list of filenames containing the data of ordinary tables,

filenames

a list of positive integers, value j at position i means that the table whose identifier is the i--th in the firstnames list is contained in the j-th file of the files component,

fusionsource

a list containing at position i the list of names of tables that store a fusion into the table whose identifier is the i--th in the firstnames list,

allnames

a list of all admissible names of ordinary library tables,

position

a list that stores at position i the position in firstnames of the identifier of the table with the i--th admissible name in allnames,

projections

a list of triples [ name , factname , map ] describing a factor fusion map from the table with identifier name to the table with identifier factname (this is used to construct the table of name using the data of the table of factname),

simpleinfo

a list of triples [ m , name , a ] describing the tables of simple groups in the library; name is the identifier of the table, m.name and name.a are admissible names for its Schur multiplier and automorphism group, respectively,

sporadicSimple

a list of identifiers of the tables of the 26 sporadic simple groups, and

GENERIC

a record with information about generic tables (see Generic Character Tables).

There are three different ways how the table data can be stored in the file.

Full ordinary tables are encoded by a call to the function MOT, where the arguments correspond to the relevant attribute values; each fusion into another library table is added by a call to ALF, values to be stored in components of the table object are added with ARC, and admissible names are notified with ALN. The argument of MOT that encodes the irreducible characters is abbreviated as follows. For each subset of characters that differ just by multiplication with a linear character or by Galois conjugacy, only the first 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 in the list) or [GALOIS,[i,j]] (which means that the character is obtained from the i-th character by applying GaloisCyc( ., j ) to it.

Brauer tables are stored relative to the corresponding ordinary tables; attribute values that can be got by restriction from the ordinary table to p--regular classes are not stored, and instead of the irreducible characters the files contain (inverses of) decomposition matrices or Brauer trees for the blocks of nonzero defects.

Ordinary construction tables have a component construction, with value a function of one argument that is called by CharacterTable when the table is constructed (not when the file containing the table is read). The aim of this mechanism is to store structured character tables such as tables of direct products and tables of central extensions of other tables in a very compact way.

2.9 How to Extend the Character Table Library

GAP users may want to extend the character table library in different respects. Probably the easiest change is to add new admissible names to library tables, in order to use these names in calls of CharacterTable (see CharacterTable in the GAP Reference Manual, and CharacterTableFromLibrary). This can be done as follows.

notifies the strings in the list newnames as new admissible names for the library table with Identifier value firstname, see Identifier!for character tables in the GAP Reference Manual. If there is already another library table for which some of these names are admissible then an error is signaled.

NotifyNameOfCharacterTable modifies the global variable LIBLIST.

ALN is a shorthand for NotifyNameOfCharacterTable. In those library files for which the maketbl script has produced the necessary information for LIBLIST, ALN is set to Ignore in the beginning and back to NotifyNameOfCharacterTable in the end.

gap> CharacterTable( "private" );
fail
gap> NotifyNameOfCharacterTable( "A5", [ "private" ] );
gap> a5:= CharacterTable( "private" );
CharacterTable( "A5" )

The next kind of changes is the addition of new fusions between library tables. Once a fusion map is known, it can be added to the library file containing the table of the subgroup, using the format produced by LibraryFusion.

ALF stores the fusion map map between the ordinary character tables with identifier strings from and to in the record encoding the table with identifier from. If the string text is given then it is added as text component of the fusion.

ALF changes the global list LIBLIST.fusionsource.

Note that the ALF statement should be placed in the file containing the data for the table with identifier from.

For a string name that is an Identifier value (see Identifier!for character tables in the GAP Reference Manual) of an ordinary character table in the GAP library, and a record fus with components name (the identifier of the destination table), map (the fusion map, a list of image positions), and optionally text (a string containing information about the fusion), LibraryFusion returns a string whose printed value can be used to add the fusion in question to the library file containing the data for the table with identifier name.

name may also be a character table, in this case its Identifier value is used as string.

gap> s5:= CharacterTable( "S5" );
CharacterTable( "A5.2" )
gap> fus:= PossibleClassFusions( a5, s5 );
[ [ 1, 2, 3, 4, 4 ] ]
gap> fusion:= rec( name:= Identifier( s5 ), map:= fus[1], text:= "unique" );;
gap> Print( LibraryFusion( "A5", fusion ) );
ALF("A5","A5.2",[1,2,3,4,4],[
"unique"
]);

The last kind of changes is the addition of new character tables to the GAP character table library. Data files containing tables in library format (i.e., in the form of calls to MOT or MBT) can be produced using PrintToLib.

prints the (ordinary or Brauer) character table tbl in library format to the file file.tbl or file (if this has already the suffix .tbl), respectively.

If tbl is an ordinary table then the value of the attribute NamesOfFusionSources is ignored by PrintToLib, since for library tables this information is extracted from the source files by the maketbl script.

gap> PrintToLib( "private", a5 );

SET_TABLEFILENAME("private");
MOT("A5",
[
"origin: ATLAS of finite groups, tests: 1.o.r., pow[2,3,5]"
],
[60,4,3,5,5],
[,[1,1,3,5,4],[1,2,1,5,4],,[1,2,3,1,1]],
[[1,1,1,1,1],[3,-1,0,-E(5)-E(5)^4,-E(5)^2-E(5)^3],
[GALOIS,[2,2]],[4,0,1,-1,-1],[5,1,-1,0,0]],
[(4,5)]);
ARC("A5","projectives",["2.A5",[[2,0,-1,E(5)+E(5)^4,E(5)^2+E(5)^3],
[GALOIS,[1,2]],[4,0,1,-1,-1],[6,0,0,1,1]],]);
ARC("A5","extInfo",["2","2"]);

If you have an ordinary character table in library format which you want to add to the table library, for example because it shall be accessible via CharacterTable (see CharacterTableFromLibrary), you must notify this table, i.e., tell GAP in which file it can be found, and which names shall be admissible for it.

notifies a new ordinary table to the library. This table has identifier component firstname, it is contained (in library format, see PrintToLib) in the file with name filename (without suffix .tbl, and relative to the pkg/ctbllib/data directory of the GAP installation), and the names contained in the list othernames are admissible for it.

NotifyCharacterTable modifies the global variable LIBLIST for the current GAP session, after having checked that there is no other library table yet with an admissible name equal to firstname or contained in othernames.

For example, let us change the name A5 to icos wherever it occurs in the file private.tbl that was produced above, and then notify the ``new'' table in this file as follows. (The name change is needed because GAP knows already a table with name A5 and would not accept to add another table with this name.)

gap> NotifyCharacterTable( "icos", "private", [] );
gap> icos:= CharacterTable( "icos" );
#W  revision entry missing in "private.tbl"
CharacterTable( "icos" )
gap> Display( icos );
icos

     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

So the private table is treated as a library table. Note that the table can be accessed only if it has been notified in the current GAP session. For frequently used private tables, it may be reasonable to put the NotifyCharacterTable statements into your .gaprc file (see The .gaprc File in the GAP Reference Manual), or into a file that is read via the .gaprc file. For adding character tables to the GAP distribution, please send the tables to the e-mail address mentioned in the first paragraph of this chapter.

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