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%W  interfac.msk           GAP documentation                Thomas Breuer
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%H  $Id: interfac.msk,v 1.18 2001/02/28 17:18:37 gap Exp $
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%Y  Copyright (C) 2001, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Chapter{The User Interface of the AtlasRep Package}

The *user interface* is the part of the {\GAP} interface that allows one
to display information about the current contents of the database and to
access individual data (perhaps from a remote server,
see~"Local or Remote Installation of the AtlasRep Share Package").
The corresponding functions are described in this chapter.
See the last section for some small examples how to use the functions of
the interface.

For the rest of the {\GAP} interface, which can be regarded as
its low level part, see Chapter~"Technicalities of the AtlasRep Package".

As stated in~"The GAP Interface to the Atlas of Group Representations",
the user interface is preliminary.
It will be extended when the {\GAP} version of the {\ATLAS} of Group
Representations is connected to other {\GAP} databases such as the
libraries of character tables and tables of marks.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Accessing vs. Constructing Representations}

Note that *accessing* the data means in particular that it is *not* the
aim of this package to construct representations from known ones.
For example, if at least one permutation representation for a group $G$
is stored but no matrix representation in a positive characteristic $p$
then the function `OneAtlasGeneratingSet' (see~"OneAtlasGeneratingSet")
returns `fail' when it is asked for a generating set for $G$ in
characteristic $p$, although such a representation can be obtained by
reduction modulo $p$ of an integral matrix representation,
which in turn can be constructed from any permutation representation.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Group Names Used in the AtlasRep Package}

The AtlasRep package refers to data of the {\ATLAS} of group representations
by the *name* of the group in question plus additional information.
Thus it is essential to know this name,
which is called *the {\GAP} name* of the group in the following.

For an almost simple group, the {\GAP} name is equal to the `Identifier'
value (see~"ref:Identifier" in the {\GAP} Reference Manual) of the
character table of this group in the {\GAP} library
(see~"ref:Access to Library Character Tables" in the {\GAP} Reference Manual);
this name is usually very similar to the name used in the
{\ATLAS} of Finite Groups (\cite{CCN85}),
for example, `\"M22\"' is the {\GAP} name of the Mathieu group $M_{22}$,
and `\"12_1.U4(3).2_1\"' is the {\GAP} name of $12_1\.U_4(3)\.2_1$.

Internally, for example as part of filenames
(see~"Filenames Used in the Atlas of Group Representations"),
the package uses names that may differ from the {\GAP} names;
these names are called *{\ATLAS}-file names*.
For example, `A5', `TE62', and `F24' are possible values for <groupname>.
Only `A5' is also a {\GAP} name, but the other two are not;
the corresponding {\GAP} names are `2E6(2)' and `Fi24{'}', respectively.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Standard Generators Used in the AtlasRep Package}

For the general definition of *standard generators* of a group,
see~"ref:Standard Generators of Groups" in the {\GAP} Reference Manual;
details can be found in~\cite{Wil96}.

Several *different* standard generators may be defined for a group,
the definitions can be found at~\URL{http://www.mat.bham.ac.uk/atlas}

When one specifies the standardization,
the $i$-th set of standard generators is denoted by the number $i$.
Note that when more than one set of standard generators is defined for
a group, one must be careful to use *compatible standardization*.
For example, the straight line programs in the database refer to a
specific standardization of their inputs.
That is, a straight line program for computing generators of a certain
subgroup of a group $G$ is defined only for a specific set of standard
generators of $G$, and applying the program to matrix or permutation
generators of $G$ but w.r.t.~a different standardization may yield
unpredictable results.
Therefore the results returned by the functions described in this chapter
contain information about the standardizations they refer to.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Class Names Used in the AtlasRep Package}

For each straight line program (see~"AtlasStraightLineProgram") that is
used to compute lists of class representatives,
it is essential to describe the classes in which these elements lie.
Therefore, in these cases the records returned by the function
`AtlasStraightLineProgram' (see~"AtlasStraightLineProgram") contain
a component `outputs' with value a list of *class names*.
Currently we define these class names only for almost simple groups,
that is, subgroups of the automorphism groups of simple groups.

For the definition of class names of an almost simple group,
we assume that the ordinary character tables of all nontrivial normal
subgroups are shown in the {\ATLAS} of Finite Groups~\cite{CCN85}.

Each class name is a string consisting of the element order of the class
in question followed by a combination of capital letters, digits, and
the characters `{'}' and `-' (starting with a capital letter).
For example, `1A', `12A1', and `3B{'}' denote the class that contains
the identity element, a class of element order $12$, and a class of
element order $3$, respectively.

\beginlist
\item{1.}
For the table of a *simple* group, the class names are the same as
returned by the {\GAP} function `ClassNames' (see~"ref:ClassNames"
in the {\GAP} Reference Manual),
see Chapter~7, Section~5 of the {\ATLAS} of Finite Groups~\cite{CCN85}:
The classes are arranged w.r.t.~increasing element order and for each
element order w.r.t.~decreasing centralizer order,
the conjugacy classes that contain elements of order $n$ are named
$nA, nB, nC, \ldots$;
the alphabet used here is potentially infinite, and reads
`A', `B', `C', $\ldots$, `Z', `A1', `B1', $\ldots$, `A2', `B2', $\ldots$.

For example, the classes of the alternating group $A_5$ have the names
`1A', `2A', `3A', `5A', and `5B'.

\item{2.}
Next we consider the case of an *upward extension* $G\.A$ of a simple
group $G$ by a *cyclic* group of order $A$.
The {\ATLAS} defines class names for each element $g$ of $G\.A$ only
w.r.t.~the group $G\.a$, say, that is generated by $G$ and $g$;
namely, there is a power of $g$ (with the exponent coprime to the order
of $g$) for which the class has a name of the same form as
the class names for simple groups,
and the name of the class of $g$ w.r.t.~$G\.a$ is then obtained from this
name by appending a suitable number of dashes~`{'}'.
So dashed class names refer exactly to those classes that are not printed
in the {\ATLAS}.

For example, those classes of the symmetric group $S_5$ that do not lie
in $A_5$ have the names `2B', `4A', and `6A'.
The outer classes of the group $L_2(8)\.3$ have the names
`3B', `6A', `9D', and `3B{'}', `6A{'}', `9D{'}'.
and the outer elements of order $5$ in the group $Sz(32)\.5$ lie in the
classes with names `5B', `5B{'}', `5B{''}', `5B{'''}', and `5B{''''}'.

In the group $G\.A$, the class of $g$ may fuse with other classes.
The name of the class of $g$ in $G\.A$ is obtained from the names of the
involved classes of $G\.a$ by concatenating their names after removing
the element order part for all of them except the first one.

For example, the elements of order $9$ in the group $L_2(27)\.6$ are
contained in the subgroup $L_2(27)\.3$ but not in $L_2(27)$.
In $L_2(27)\.3$, they lie in the classes `9A', `9A{'}', `9B', and `9B{'}';
in $L_2(27)\.6$, these classes fuse to `9AB' and `9A{'}B{'}'.

\item{3.}
Now we define class names for *general upward extensions* $G\.A$ of a
simple group.
Each element $g$ of such a group lies in an upward extension $G\.a$ by a
cyclic group, and the class names w.r.t.~$G\.a$ are already defined.
The name of the class of $g$ in $G\.A$ is obtained by concatenating the
names of the classes in the orbit of $G\.A$ on the classes of cyclic
upward extensions of $G$, after ordering the names lexicographically and
removing the element order part for all of them except the first one.
An *exception* is the situation where dashed and non-dashed class names
appear in an orbit; in this case, the dashed names are omitted.

For example, the classes `21A' and `21B' of the group $U_3(5)\.3$ fuse
in $U_3(5)\.S_3$ to the class `21AB',
and the class `2B' of $U_3(5)\.2$ fuses with the involution classes
`2B{'}', `2B{''}' in the groups $U_3(5)\.2^{\prime}$ and
$U_3(5)\.2^{\prime\prime}$ to the class `2B' of $U_3(5)\.S_3$.
\endlist

It may happen that some names in the `outputs' component of a record
returned by `AtlasStraightLineProgram' (see~"AtlasStraightLineProgram")
do not uniquely determine the classes of the corresponding elements.
For example, the (algebraically conjugate) classes `39A' and `39B' of the
group $Co_1$ have not been distinguished yet.
In such cases, the names used contain a minus sign `-',
and mean ``one of the classes in the range described by the name before
and the name after the minus sign''; the element order part of the name
does not appear after the minus sign.
So the name `39A-B' for the group $Co_1$ means `39A' or `39B',
and the name `20A-B{''''}' for the group $Sz(32)$ means one of the
(outer) classes of element order $20$ in this group.



\>AtlasClassNames( <tbl> ) F

If <tbl> is the ordinary character table of an almost simple group $G$
whose ordinary character table is contained in the {\ATLAS} of Finite
Groups then `AtlasClassNames' returns the list of class names for $G$,
as defined above.
The ordering of class names is the same as the ordering of the columns
of <tbl>.

(The function may work also for almost simple non-{\ATLAS} groups,
but then clearly the class names returned are somewhat arbitrary.)



\beginexample 
gap> AtlasClassNames( CharacterTable( "L3(4).3" ) );
[ "1A", "2A", "3A", "4ABC", "5A", "5B", "7A", "7B", "3B", "3B'", "3C", "3C'", 
  "6B", "6B'", "15A", "15A'", "15B", "15B'", "21A", "21A'", "21B", "21B'" ]
gap> AtlasClassNames( CharacterTable( "U3(5).2" ) );
[ "1A", "2A", "3A", "4A", "5A", "5B", "5CD", "6A", "7AB", "8AB", "10A", "2B", 
  "4B", "6D", "8C", "10B", "12B", "20A", "20B" ]
gap> AtlasClassNames( CharacterTable( "L2(27).6" ) );
[ "1A", "2A", "3AB", "7ABC", "13ABC", "13DEF", "14ABC", "2B", "4A", "26ABC", 
  "26DEF", "28ABC", "28DEF", "3C", "3C'", "6A", "6A'", "9AB", "9A'B'", "6B", 
  "6B'", "12A", "12A'" ]
\endexample


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Accessing Data of the AtlasRep Package}

(Note that the output of the examples in this section refers to a perhaps
outdated table of contents;
the current version of the database may contain more information
than is shown here.)

\>DisplayAtlasInfo(  ) F
\>DisplayAtlasInfo( <listofnames> ) F
\>DisplayAtlasInfo( <gapname>[, <std>] ) F
\>DisplayAtlasInfo( <gapname>[, <std>], IsPermGroup ) F
\>DisplayAtlasInfo( <gapname>[, <std>], NrMovedPoints, <n> ) F
\>DisplayAtlasInfo( <gapname>[, <std>], IsMatrixGroup ) F
\>DisplayAtlasInfo( <gapname>[, <std>][, Characteristic, <p>]
[, Dimension, <n>] ) F
\>DisplayAtlasInfo( <gapname>[, <std>], IsStraightLineProgram ) F

`DisplayAtlasInfo' lists the information available via the AtlasRep
package, for the given inputs.
Depending on the value of the `remote' component of the global variable
`AtlasOfGroupRepresentationsInfo'
(see~"AtlasOfGroupRepresentationsInfo"), all the data provided by the
{\ATLAS} of Group Representations or only that in the local installation
is available.

Called without arguments, `DisplayAtlasInfo' prints an overview what
information the {\ATLAS} of Group Representations provides.
One line is printed for each group $G$, with the following columns.
\beginitems
`group' &
    the {\GAP} name of $G$
    (see~"Group Names Used in the AtlasRep Package"),

`\#' &
    the number of representations stored for $G$,

`maxes' &
    the available straight line programs\index{straight line program}
    for computing maximal subgroups of $G$,

`cl' &
    a `+' sign if at least one program for computing representatives
    of conjugacy classes of elements of $G$ is stored,
    and a `-' sign otherwise,

`cyc' &
    a `+' sign if at least one program for computing representatives of
    classes of maximally cyclic subgroups of $G$ is stored,
    and a `-' sign otherwise, and

`out' &
    descriptions of outer automorphisms of $G$ for which at least one
    program is stored.
\enditems

In the second form, <listofnames> must be a list of strings that are
{\GAP} names for a group from the {\ATLAS} of Group Representations;
in this case, the overview described above is restricted to the groups
in this list.

In the third form, <gapname> must be a string that is a {\GAP} name
for a group from the {\ATLAS} of Group Representations;
in this case, `DisplayAtlasInfo' prints an overview of the information
that is available for this group.
One line is printed for each representation, showing the number of this
representation
(which is used in calls of `AtlasGenerators', see~"AtlasGenerators"),
and a string of one of the following forms.
`G \<= Sym(<n><id>)' denotes a permutation representation of degree <n>,
and `G \<= GL(<n><id>,<descr>)' denotes a matrix representation of
dimension <n> over a coefficient ring described by <descr>;
in both cases, <id> is a (possible empty) string specifying the
representation, for example `G \<= Sym(40a)' and `G \<= Sym(40b)' denote
two (nonequivalent) representations of degree $40$.
In the case of matrix representations, <descr> can be a prime power,
`Z' (denoting the ring of integers),
a description of an algebraic extension field,
`C' (denoting an unspecified algebraic extension field), or
`Z/<m>Z' for an integer <m> (denoting the ring of residues mod <m>);
for example, `G \<= GL(2a,4)' and `G \<= GL(2b,4)' denote two
(nonequivalent) representations of dimension $2$ over the field with four
elements.
After the representations, the straight line programs available for
<gapname> are listed.

If the first argument is a string <gapname>, the following optional
arguments can be used to restrict the overview.
\beginitems
<std> &
    must be a positive integer or a list of positive integers;
    if it is given then only those representations are considered
    that refer to the <std>-th set of standard generators or the $i$-th
    set of standard generators, for $i$ in <std>
    (see~"Standard Generators Used in the AtlasRep Package"),

`IsPermGroup' &
    restricts to permutation representations,

`NrMovedPoints' and <n> &
    for a positive integer or a list <n>, restrict to permutation
    representations of degree <n> or in the list <n>,

`IsMatrixGroup' &
    restricts to matrix representations,

`Characteristic' and <p> &
    for a prime integer or a list <p>, restrict to matrix representations
    over fields of characteristic <p> or in the list <p>
    (representations over residue class rings that are not fields can be
    addressed by `<p> = fail'),

`Dimension' and <n> &
    for a positive integer <n> or a list <n>, restrict to matrix
    representations of dimension <n> or in the list <n>, and

`IsStraightLineProgram' &
    restricts to straight line programs.
\enditems

The representations are ordered as follows.
Permutation representations come first (ordered according to the degree),
followed by matrix representations over finite fields
(ordered first according to the characteristic and second according to
the dimension), matrix representations over the integers,
and then matrix representations over algebraic extension fields
(both kinds ordered according to the dimension),
the last representations are matrix representations over residue class
rings (ordered first according to the modulus and second according to the
dimension).

Note that in each case, either the whole contents of the database
or the contents of the local installation is considered,
depending on whether the value of
`AtlasOfGroupRepresentationsInfo.remote' is `true' or `false'
(see~"AtlasOfGroupRepresentationsInfo").



\beginexample 
gap> DisplayAtlasInfo( [ "M11", "A5" ] );
group     #     maxes  cl  cyc  out
-----------------------------------
M11      42      1..5   +    +     
A5       18      1..3   -    -     
\endexample

The above output means that the {\ATLAS} of Group Representations contains
$42$ representations of the Mathieu group $M_{11}$,
straight line programs for computing generators of representatives
of all five classes of maximal subgroups,
for computing representatives of the conjugacy classes of elements
and of generators of maximally cyclic subgroups,
and contains no straight line program for applying outer automorphisms
(well, in fact $M_{11}$ admits no nontrivial outer automorphism).
Analogously,
$18$ representations of the alternating group $A_5$ are available,
straight line programs for computing generators of representatives
of all three classes of maximal subgroups,
and no straight line programs for computing representatives
of the conjugacy classes of elements,
of generators of maximally cyclic subgroups,
and no for computing images under outer automorphisms.

\beginexample 
gap> DisplayAtlasInfo( "A5", IsPermGroup );
Representations for G = A5:    (all refer to std. generators 1)
---------------------------
 1: G <= Sym(5)
 2: G <= Sym(6)
 3: G <= Sym(10)
gap> DisplayAtlasInfo( "A5", NrMovedPoints, [ 4 .. 9 ] );
Representations for G = A5:    (all refer to std. generators 1)
---------------------------
 1: G <= Sym(5)
 2: G <= Sym(6)
\endexample

The first three representations stored for $A_5$ are (in fact primitive)
permutation representations.

\beginexample 
gap> DisplayAtlasInfo( "A5", Dimension, [ 1 .. 3 ] );
Representations for G = A5:    (all refer to std. generators 1)
---------------------------
 8: G <= GL(2a,4)
 9: G <= GL(2b,4)
10: G <= GL(3,5)
12: G <= GL(3a,9)
13: G <= GL(3b,9)
17: G <= GL(3a,Field([Sqrt(5)]))
18: G <= GL(3b,Field([Sqrt(5)]))
gap> DisplayAtlasInfo( "A5", Characteristic, 0 );
Representations for G = A5:    (all refer to std. generators 1)
---------------------------
14: G <= GL(4,Z)
15: G <= GL(5,Z)
16: G <= GL(6,Z)
17: G <= GL(3a,Field([Sqrt(5)]))
18: G <= GL(3b,Field([Sqrt(5)]))
\endexample

The representations with number between $4$ and $13$ are
(in fact irreducible) matrix representations over various finite fields,
those with numbers $14$ to $16$ are integral matrix representations,
and the last two are matrix representations over the field generated by
$\sqrt{5}$ over the rational number field.

\beginexample 
gap> DisplayAtlasInfo( "A5", IsStraightLineProgram );
Straight line programs for G = A5:    (all refer to std. generators 1)
----------------------------------
available maxes of G:  [ 1 .. 3 ]
repres. of cyclic subgroups of G available?  false
class repres. of G available?  false
available automorphisms:  [  ]
\endexample

Straight line programs are available to compute generators of representatives
of the three classes of maximal subgroups of $A_5$,
see~"AtlasStraightLineProgram".

\>AtlasGenerators( <gapname>, <repnr> ) F
\>AtlasGenerators( <gapname>, <repnr>, <maxnr> ) F
\>AtlasGenerators( <identifier> ) F

In the first two forms, <gapname> must be a string denoting a {\GAP}
name (see~"Group Names Used in the AtlasRep Package") of a group,
and <repnr> a positive integer.
If the {\ATLAS} of Group Representations contains at least <repnr>
representations for the group with {\GAP} name <gapname> then
`AtlasGenerators', when called with <gapname> and <repnr>,
returns an immutable record describing the <repnr>-th representation;
otherwise `fail' is returned.
If a third argument <maxnr>, a positive integer,
is given then an immutable record describing the restriction of the
<repnr>-th representation to the <maxnr>-th maximal subgroup is returned.

The result record has the following components.
\beginitems
`generators' &
    a list of generators for the group,

`standardization' &
    the positive integer denoting the underlying standard generators,

`identifier' &
    a {\GAP} object (a list of filenames plus additional information)
    that uniquely determines the representation;
    the value can be used as <identifier> argument of `AtlasGenerators'
    (see below).
\enditems
It should be noted that the number <repnr> refers to the number shown
by `DisplayAtlasInfo' (see~"DisplayAtlasInfo") *in the current session*;
it may be that after the addition of new representations, <repnr> refers
to another representation.

The third form of `AtlasGenerators', with only argument <identifier>,
can be used to fetch the result record with `identifier' value equal to
<identifier>.
The purpose of this variant is to access the *same* representation also
in *different* {\GAP} sessions.



\beginexample 
gap> gens1:= AtlasGenerators( "A5", 1 );
rec( generators := [ (1,2)(3,4), (1,3,5) ], standardization := 1, 
  identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ] )
gap> gens8:= AtlasGenerators( "A5", 8 );
rec( 
  generators := [ [ [ Z(2)^0, 0*Z(2) ], [ Z(2^2), Z(2)^0 ] ], [ [ 0*Z(2), Z(2
                 )^0 ], [ Z(2)^0, Z(2)^0 ] ] ], standardization := 1, 
  identifier := [ "A5", [ "A5G1-f4r2aB0.m1", "A5G1-f4r2aB0.m2" ], 1, 4 ] )
gap> gens17:= AtlasGenerators( "A5", 17 );
rec( 
  generators := [ [ [ -1, 0, 0 ], [ 0, -1, 0 ], [ -E(5)-E(5)^4, -E(5)-E(5)^4, 
              1 ] ], [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ], 
  standardization := 1, identifier := [ "A5", "A5G1-Ar3aB0.M", 1, 3 ] )
\endexample

Each of the above pairs of elements generates a group isomorphic to $A_5$.

\beginexample 
gap> gens1max2:= AtlasGenerators( "A5", 1, 2 );
rec( generators := [ (1,2)(3,4), (2,3)(4,5) ], standardization := 1, 
  identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5, 2 ] )
gap> id:= gens1max2.identifier;;
gap> gens1max2 = AtlasGenerators( id );
true
gap> max2:= Group( gens1max2.generators );;
gap> Size( max2 );
10
gap> IdGroup( max2 ) = IdGroup( DihedralGroup( 10 ) );
true
\endexample

The elements stored in `gens1max2.generators' describe the restriction of
the first representation of $A_5$ to a group in the second class of
maximal subgroups of $A_5$ according to the list in the {\ATLAS} of Finite
Groups~\cite{CCN85};
this subgroup is isomorphic to the dihedral group $D_{10}$.


\>AtlasStraightLineProgram( <gapname>[, <std>], <maxnr> ) F
\>AtlasStraightLineProgram( <gapname>[, <std>], "classes" ) F
\>AtlasStraightLineProgram( <gapname>[, <std>], "cyclic" ) F
\>AtlasStraightLineProgram( <gapname>[, <std>], "automorphism", <autname> ) F
\>AtlasStraightLineProgram( <identifier> ) F

In the first four forms, <gapname> must be a string denoting a {\GAP}
name (see~"Group Names Used in the AtlasRep Package") of a group $G$,
say, and <std> a positive integer denoting the standard generators of $G$
used (see~"Standard Generators Used in the AtlasRep Package").
If the {\ATLAS} of Group Representations contains a straight line program
(see~"ref:Straight Line Programs" in the {\GAP} Reference Manual)
as described by the last argument (see below) then
`AtlasStraightLineProgram' returns an immutable record with this program,
otherwise `fail' is returned.

If the optional argument <std> is given, only those straight line
programs are considered that take generators from the <std>-th set of
standard generators of $G$ as input.

The result record has the following components.
\beginitems
`program' &
    the required straight line program,

`standardization' &
    the positive integer denoting the underlying standard generators of
    $G$,

`identifier' &
    a {\GAP} object (a list of filenames plus additional information)
    that uniquely determines the program;
    the value can be used as <identifier> argument of
    `AtlasStraightLineProgram' (see below).
\enditems

In the first form, the last argument must be a positive integer <maxnr>,
and the required program computes generators of the <maxnr>-th maximal
subgroup of the group with {\GAP} name <gapname>.%
\index{straight line program!for maximal subgroups}%
\index{maximal subgroups}

In the second form, the last argument is the string `\"classes\"',
and the required program computes representatives of conjugacy classes of
elements of the group with {\GAP} name <gapname>.%
\index{straight line program!for class representatives}%
\index{class representatives}

In the third form, the last argument is the string `\"cyclic\"',
and the required program computes representatives of generators of
classes of maximally cyclic subgroups of the group with {\GAP} name
<gapname>.%
\index{straight line program!for representatives of cyclic subgroups}%
\index{cyclic subgroups}%
\index{maximally cyclic subgroups}

See~\cite{BSW01} for the background concerning these straight line
programs.
In the second and third form of `AtlasStraightLineProgram',
the result record also contains a component `outputs',
whose value is a list of class names of the outputs,
as described in Section~"Class Names Used in the AtlasRep Package".

In the fourth form, the last argument is a string <autname>,
and the required program computes images of standard generators under the
outer automorphism of the group with {\GAP} name <gapname>
that is given by this string.
\index{straight line program!for outer automorphisms}%
\index{automorphisms}

The fifth form of `AtlasStraightLineProgram',
with only argument <identifier>,
can be used to fetch the result record with `identifier' value equal to
<identifier>.



\beginexample 
gap> prog:= AtlasStraightLineProgram( "A5", 2 );
rec( program := <straight line program>, standardization := 1, 
  identifier := [ "A5", "A5G1-max2W1", 1 ] )
gap> StringOfResultOfStraightLineProgram( prog.program, [ "a", "b" ] );
"[ a, bbab ]"
gap> gens1:= AtlasGenerators( "A5", 1 );
rec( generators := [ (1,2)(3,4), (1,3,5) ], standardization := 1, 
  identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ] )
gap> maxgens:= ResultOfStraightLineProgram( prog.program, gens1.generators );
[ (1,2)(3,4), (2,3)(4,5) ]
gap> maxgens = gens1max2.generators;
true
\endexample

The above example shows that for restricting representations given by
standard generators to a maximal subgroup of $A_5$,
we can also fetch and apply the appropriate straight line program.
Such a program (see~"ref:Straight Line Programs" in the {\GAP} Reference
Manual) takes standard generators of a group --in this example $A_5$--
as its input, and returns a list of elements in this group
--in this example generators of the $D_{10}$ subgroup we had met above--
which are computed essentially by evaluating structured words in terms of
the standard generators.

% also here: A5 example if available ...
\beginexample 
gap> prog:= AtlasStraightLineProgram( "J1", "cyclic" );
rec( program := <straight line program>, standardization := 1, 
  identifier := [ "J1", "J1G1-cycW1", 1 ],
  outputs := [ "6A", "7A", "10B", "11A", "15B", "19A" ] )
gap> gens:= GeneratorsOfGroup( FreeGroup( "x", "y" ) );;
gap> ResultOfStraightLineProgram( prog.program, gens );
[ x*y*x*y^2*x*y*x*y^2*x*y*x*y*x*y^2*x*y^2, x*y, x*y*x*y^2*x*y*x*y*x*y^2*x*y^2,
  x*y*x*y*x*y^2*x*y^2*x*y*x*y^2*x*y*x*y*x*y^2*x*y^2*x*y*x*y^2*x*y*x*y*x*y^
    2*x*y^2, x*y*x*y*x*y^2*x*y^2, x*y*x*y^2 ]
\endexample

The above example shows how to fetch and use straight line programs for
computing generators of representatives of maximally cyclic subgroups
of a given group.

\>OneAtlasGeneratingSet( <gapname>[, <std>] ) F
\>OneAtlasGeneratingSet( <gapname>[, <std>], IsPermGroup ) F
\>OneAtlasGeneratingSet( <gapname>[, <std>], NrMovedPoints, <n> ) F
\>OneAtlasGeneratingSet( <gapname>[, <std>], IsMatrixGroup ) F
\>OneAtlasGeneratingSet( <gapname>[, <std>][, Characteristic, <p>]
[, Dimension, <m>] ) F
\>OneAtlasGeneratingSet( <gapname>[, <std>][, Ring, <R>]
[, Dimension, <m>] ) F

Let <gapname> be a string.
If the {\ATLAS} of Group Representations contains at least one
representation for the group $G$ with {\GAP} name <gapname> and with
the required properties then `OneAtlasRepresentation' returns
an immutable record as described for `AtlasGenerators'
(see~"AtlasGenerators") for such a representation;
otherwise `fail' is returned.
If the argument <std> is given then it must be a positive integer
or a list of positive integers, denoting the sets of standard generators
w.r.t.~which the representation shall be given
(see~"Standard Generators Used in the AtlasRep Package").

In the first form, `OneAtlasGeneratingSet' returns the first generating
set for $G$, in the ordering shown by `DisplayAtlasInfo'
(see~"DisplayAtlasInfo").
In the remaining forms,
`IsPermGroup' and `IsMatrixGroup' restrict to permutation and matrix
representations, respectively,
`NrMovedPoints' restricts to permutation representations of degree equal
to the integer <n> or in the list <n>,
`Characteristic' restricts to matrix representations of characteristic
equal to the integer <p> or in the list <p> (representations over residue
class rings that are not fields can be accessed with `<p> = fail'),
`Dimension' restricts to matrix representations of dimension equal to the
positive integer <m> or in the list <m>,
and `Ring' restricts to matrix representations with matrix entries
contained in the ring <R>.



First we access a permutation representation for the alternating group $A_5$.

\beginexample 
gap> gens:= OneAtlasGeneratingSet( "A5" );
rec( generators := [ (1,2)(3,4), (1,3,5) ], standardization := 1, 
  identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ] )
gap> gens = OneAtlasGeneratingSet( "A5", IsPermGroup );
true
gap> gens = OneAtlasGeneratingSet( "A5", NrMovedPoints, [ 1 .. 10 ] );
true
gap> OneAtlasGeneratingSet( "A5", NrMovedPoints, 20 );
fail
\endexample

Note that a permutation representation of degree $20$ could be obtained
by taking twice the primitive representation on $10$ points;
however, the {\ATLAS} of Group Representations does not store this
representation (cf.~"Accessing vs. Constructing Representations").

Next we access matrix representations of $A_5$.

\beginexample 
gap> gens:= OneAtlasGeneratingSet( "A5", IsMatrixGroup );
rec( 
  generators := [ <an immutable 4x4 matrix over GF2>, <an immutable 4x4 matrix\
 over GF2> ], standardization := 1, 
  identifier := [ "A5", [ "A5G1-f2r4aB0.m1", "A5G1-f2r4aB0.m2" ], 1, 2 ] )
gap> gens = OneAtlasGeneratingSet( "A5", Dimension, 4 );
true
gap> gens = OneAtlasGeneratingSet( "A5", Characteristic, 2 );
true
gap> OneAtlasGeneratingSet( "A5", Characteristic, [2,5], Dimension, 2 );
rec( 
  generators := [ [ [ Z(2)^0, 0*Z(2) ], [ Z(2^2), Z(2)^0 ] ], [ [ 0*Z(2), Z(2
                 )^0 ], [ Z(2)^0, Z(2)^0 ] ] ], standardization := 1, 
  identifier := [ "A5", [ "A5G1-f4r2aB0.m1", "A5G1-f4r2aB0.m2" ], 1, 4 ] )
gap> OneAtlasGeneratingSet( "A5", Characteristic, [2,5], Dimension, 1 );
fail
gap> gens:= OneAtlasGeneratingSet( "A5", Characteristic, 0, Dimension, 4 );
rec( 
  generators := [ [ [ 1, 0, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ], [ -1, -1, 
              -1, -1 ] ], 
      [ [ 0, 1, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 1, 0, 0, 0 ] ] ], 
  standardization := 1, identifier := [ "A5", "A5G1-Zr4B0.M", 1, 4 ] )
gap> gens = OneAtlasGeneratingSet( "A5", Ring, Integers );
true
gap> gens = OneAtlasGeneratingSet( "A5", Ring, CF(37) );
true
gap> OneAtlasGeneratingSet( "A5", Ring, Integers mod 77 );
fail
gap> gens:= OneAtlasGeneratingSet( "A5", Ring, CF(5), Dimension, 3 );
rec( 
  generators := [ [ [ -1, 0, 0 ], [ 0, -1, 0 ], [ -E(5)-E(5)^4, -E(5)-E(5)^4, 
              1 ] ], [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ], 
  standardization := 1, identifier := [ "A5", "A5G1-Ar3aB0.M", 1, 3 ] )
gap> OneAtlasGeneratingSet( "A5", Ring, GF(17) );
fail
\endexample

Note that a matrix representation in any characteristic can be obtained by
reducing a permutation representation or an integral matrix representation;
however, the {\ATLAS} of Group Representations does not store such a
representation (cf.~"Accessing vs. Constructing Representations").


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Updating the Tables of Contents of the AtlasRep Package}

\>ReloadAtlasTableOfContents( \"local\" ) F
\>ReloadAtlasTableOfContents( \"remote\" ) F

`ReloadAtlasTableOfContents' replaces the table of contents
(local or remote, depending on the argument)
of the {\ATLAS} of Group Representations by the one obtained from
inspecting the actual contents of the data directories
(see~"The Tables Of Contents of the Atlas of Group Representations").




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Examples of Using the AtlasRep Package}

*Example~1: Class Representatives*

First we show the computation of class representatives of a $2$-modular
representation of the Mathieu group $M_{11}$.
We start with the ordinary and Brauer character tables of this group.

\beginexample 
gap> tbl:= CharacterTable( "M11" );;
gap> modtbl:= tbl mod 2;;
gap> CharacterDegrees( modtbl );
[ [ 1, 1 ], [ 10, 1 ], [ 16, 2 ], [ 44, 1 ] ]
\endexample

The output of `CharacterDegrees' (see~"ref:CharacterDegrees" in the
{\GAP} Reference Manual) means that the $2$-modular irreducibles of $M_{11}$
have degrees $1$, $10$, $16$, $16$, and $44$.

We are interested in the irreducible $10$-dimensional representation;
using `DisplayAtlasInfo', we find out that matrix generators for this
representation are available in the AtlasRep database.

\beginexample 
gap> DisplayAtlasInfo( "M11", Characteristic, 2 );
Representations for G = M11:    (all refer to std. generators 1)
----------------------------
 6: G <= GL(10,2)
 7: G <= GL(32,2)
 8: G <= GL(44,2)
16: G <= GL(16b,4)
17: G <= GL(16a,4)
\endexample

We fetch the generators, and compute the list of class representatives
of $M_{11}$ in the representation.
The ordering of class representatives is the same as that in the character
table of the {\ATLAS} of Finite Groups (\cite{CCN85}),
which coincides with the ordering of columns in the {\GAP} table we have
fetched above.

\beginexample 
gap> gens:= OneAtlasGeneratingSet( "M11", Characteristic, 2,
>                                         Dimension, 10 );;
gap> ccls:= AtlasStraightLineProgram( "M11", gens.standardization,
>                                     "classes" );
rec( program := <straight line program>, standardization := 1,
  identifier := [ "M11", [ "M11G1cycW1-cclsW1", "M11G1-cycW1" ], 1 ],
  outputs := [ "1A", "2A", "3A", "4A", "5A", "6A", "8A", "8B", "11A", "11B" ]
 )
gap> reps:= ResultOfStraightLineProgram( ccls.program, gens.generators );;
\endexample

Let us check whether the class representatives have the right orders.

\beginexample 
gap> Length( reps );
10
gap> List( reps, Order ) = OrdersClassRepresentatives( tbl );
true
\endexample

{}From the class representatives, we can compute the Brauer character
we had started with.
This Brauer character is defined on all classes of the $2$-modular table.
So we first pick only those representatives,
using the {\GAP} function `GetFusionMap'
(see~"ref:GetFusionMap" in the {\GAP} Reference Manual);
in this situation, it returns the class fusion from the Brauer table into
the ordinary table.

\beginexample 
gap> fus:= GetFusionMap( modtbl, tbl );
[ 1, 3, 5, 9, 10 ]
gap> modreps:= reps{ fus };;
\endexample

Then we call the {\GAP} function `BrauerCharacterValue'
(see~"ref:BrauerCharacterValue" in the {\GAP} Reference Manual),
which computes the Brauer character value from the matrix given.

\beginexample 
gap> char:= List( modreps, BrauerCharacterValue );
[ 10, 1, 0, -1, -1 ]
gap> Position( Irr( modtbl ), char );
2
\endexample


*Example~2: Permutation and Matrix Representations*

The second example shows the computation of a permutation representation
from a matrix representation.
We work with the $10$-dimensional representation used above,
and consider the action on the $2^{10}$ vectors of the underlying row space.

\beginexample 
gap> grp:= Group( gens.generators );;
gap> v:= GF(2)^10;;
gap> orbs:= Orbits( grp, AsList( v ) );;
gap> List( orbs, Length );
[ 1, 396, 55, 330, 66, 165, 11 ]
\endexample

We see that there are six nontrivial orbits, and we can compute the
permutation actions on these orbits directly using `Action'.
However, for larger examples, one cannot write down all orbits on the
row space, so one has to use another strategy if one is interested in
a particular orbit.

Let us assume that we are interested in the orbit of length $11$.
The point stabilizer is the first maximal subgroup of $M_{11}$,
thus the restriction of the representation to this subgroup has a
nontrivial fixed point space.
This restriction can be computed using the AtlasRep package.

\beginexample 
gap> gens:= AtlasGenerators( "M11", 6, 1 );;
\endexample

Now computing the fixed point space is standard linear algebra.

\beginexample 
gap> id:= IdentityMat( 10, GF(2) );;
gap> sub1:= Subspace( v, NullspaceMat( gens.generators[1] - id ) );;
gap> sub2:= Subspace( v, NullspaceMat( gens.generators[2] - id ) );;
gap> fix:= Intersection( sub1, sub2 );
<vector space of dimension 1 over GF(2)>
\endexample

The final step is of course the computation of the permutation action
on the orbit.

\beginexample 
gap> orb:= Orbit( grp, Basis( fix )[1] );;
gap> act:= Action( grp, orb );;  Print( act, "\n" );
Group( [ ( 1, 2)( 4, 6)( 5, 8)( 7,10), ( 1, 3, 5, 9)( 2, 4, 7,11) ] )
\endexample

Note that this group is *not* equal to the group obtained by fetching
the permutation representation from the database.
This is due to a different numbering of the points,
so the groups are permutation isomorphic.

\beginexample 
gap> permgrp:= Group( AtlasGenerators( "M11", 1 ).generators );;
gap> Print( permgrp, "\n" );
Group( [ ( 2,10)( 4,11)( 5, 7)( 8, 9), ( 1, 4, 3, 8)( 2, 5, 6, 9) ] )
gap> permgrp = act;
false
gap> IsConjugate( SymmetricGroup(11), permgrp, act );
true
\endexample


*Example~3: Outer Automorphisms*

The straight line programs for applying outer automorphisms to
standard generators can of course be used to define the automorphisms
themselves as {\GAP} mappings.

\beginexample 
gap> DisplayAtlasInfo( "G2(3)", IsStraightLineProgram );
Straight line programs for G = G2(3):    (all refer to std. generators 1)
-------------------------------------
available maxes of G:  [ 1 .. 10 ]
repres. of cyclic subgroups of G available?  true
class repres. of G available?  false
available automorphisms:  [ "2" ]
gap> prog:= AtlasStraightLineProgram( "G2(3)", "2" ).program;;
gap> gens:= AtlasGenerators( "G2(3)", 1 ).generators;;
gap> imgs:= ResultOfStraightLineProgram( prog, gens );;
gap> g:= Group( gens );;
\endexample

If we are not suspicious whether the script really describes an
automorphism then we should tell this to {\GAP},
in order to avoid the expensive checks of the properties of being a
homomorphism and bijective (see~"ref:Creating Group Homomorphisms"
in the {\GAP} Reference Manual).
This looks as follows.

\beginexample 
gap> aut:= GroupHomomorphismByImagesNC( g, g, gens, imgs );;
gap> SetIsBijective( aut, true );
\endexample

If we are suspicious whether the script describes an automorphism
then we might have the idea to check it with {\GAP}, as follows.

\begintt
gap> aut:= GroupHomomorphismByImages( g, g, gens, imgs );;
gap> IsBijective( aut );
true
\endtt

(However, even for a comparatively small group such as $G_2(3)$,
this is a difficult task, at least for {\GAP}~4.2;
in {\GAP}~4.3, it will be easier.)

Often one can form images under an automorphism $\alpha$, say,
without creating the homomorphism object.
This is obvious for the standard generators of the group $G$ themselves,
but also for generators of a maximal subgroup $M$ computed from standard
generators of $G$, provided that the straight line programs in question
refer to the same standard generators.
Note that the generators of $M$ are given by evaluating words in terms of
standard generators of $G$,
and their images under $\alpha$ can be obtained by evaluating the same words
at the images under $\alpha$ of the standard generators of $G$.

\beginexample 
gap> max1:= AtlasStraightLineProgram( "G2(3)", 1 ).program;;
gap> mgens:= ResultOfStraightLineProgram( max1, gens );;
gap> comp:= CompositionOfStraightLinePrograms( max1, prog );;
gap> mimgs:= ResultOfStraightLineProgram( comp, gens );;
\endexample

The list `mgens' is the list of generators of the first maximal subgroup
of $G_2(3)$, `mimgs' is the list of images under the automorphism given
by the straight line program `prog'.
Note that applying the program returned by
`CompositionOfStraightLinePrograms' means to apply first `prog' and then
`max1'.
(From version 4.3 of {\GAP} on,
the function `CompositionOfStraightLinePrograms' is documented in the
{\GAP} Reference Manual.)
 
Since we have already constructed the {\GAP} object representing the
automorphism, we can check whether the results are equal.

\beginexample 
gap> mimgs = List( mgens, x -> x^aut );
true
\endexample

However, it should be emphasized that using `aut' requires a huge
machinery of computations behind the scenes, whereas applying the
straight line programs `prog' and `max1' involves only elementary
operations with the generators.
The latter is feasible also for larger groups,
for which constructing the {\GAP} automorphism might be impossible.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Section{The Atlas as Part of the GAP Database of Groups}
%
% missing ...
%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%E

