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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
Project: cocalc-sagemath-dev-slelievre
Views: 418346#############################################################################
##
#W interfac.gd GAP 4 package AtlasRep Thomas Breuer
##
#Y Copyright (C) 2001, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany
##
## This file contains the declaration part of the ``high level'' GAP
## interface to the ATLAS of Group Representations.
##
#############################################################################
##
#F DisplayAtlasInfo( [<listofnames>][,][<std>][,]["contents", <sources>]
#F [, IsPermGroup[, true]]
#F [, NrMovedPoints, <n>]
#F [, IsMatrixGroup[, true]]
#F [, Characteristic, <p>][, Dimension, <n>]
#F [, Position, <n>]
#F [, Character, <chi>]
#F [, Identifier, <id>] )
#F DisplayAtlasInfo( <gapname>[, <std>][, "contents", <sources>]
#F [, IsPermGroup[, true]]
#F [, NrMovedPoints, <n>]
#F [, IsMatrixGroup[, true]]
#F [, Characteristic, <p>][, Dimension, <n>]
#F [, Position, <n>]
#F [, Character, <chi>]
#F [, Identifier, <id>]
#F [, IsStraightLineProgram[, true]] )
##
## <#GAPDoc Label="DisplayAtlasInfo">
## <ManSection>
## <Func Name="DisplayAtlasInfo"
## Arg='[listofnames][,][std][,]["contents", sources][, ...]'/>
## <Func Name="DisplayAtlasInfo" Arg='gapname[, std][, ...]'
## Label="for a group name, and optionally further restrictions"/>
##
## <Description>
## This function lists the information available via the
## <Package>AtlasRep</Package> package, for the given input.
## Depending on whether remote access to data is enabled
## (see Section <Ref Subsect="subsect:Local or remote access"/>),
## all the data provided by the &ATLAS; of Group Representations
## or only those in the local installation are considered.
## <P/>
## An interactive alternative to <Ref Func="DisplayAtlasInfo"/> is the
## function <Ref Func="BrowseAtlasInfo" BookName="Browse"/>,
## see <Cite Key="Browse"/>.
## <P/>
## Called without arguments, <Ref Func="DisplayAtlasInfo"/> prints an
## overview what information the &ATLAS; of Group Representations provides.
## One line is printed for each group <M>G</M>, with the following columns.
## <P/>
## <List>
## <Mark><C>group</C></Mark>
## <Item>
## the &GAP; name of <M>G</M> (see
## Section <Ref Sect="sect:Group Names Used in the AtlasRep Package"/>),
## </Item>
## <Mark><C>#</C></Mark>
## <Item>
## the number of faithful representations stored for <M>G</M>
## that satisfy the additional conditions given (see below),
## </Item>
## <Mark><C>maxes</C></Mark>
## <Item>
## the number of available straight line programs
## <Index>straight line program</Index>
## for computing generators of maximal subgroups of <M>G</M>,
## </Item>
## <Mark><C>cl</C></Mark>
## <Item>
## a <C>+</C> sign if at least one program for computing representatives
## of conjugacy classes of elements of <M>G</M> is stored,
## </Item>
## <Mark><C>cyc</C></Mark>
## <Item>
## a <C>+</C> sign if at least one program for computing representatives
## of classes of maximally cyclic subgroups of <M>G</M> is stored,
## </Item>
## <Mark><C>out</C></Mark>
## <Item>
## descriptions of outer automorphisms of <M>G</M> for which at least
## one program is stored,
## </Item>
## <Mark><C>fnd</C></Mark>
## <Item>
## a <C>+</C> sign if at least one program is available for finding
## standard generators,
## </Item>
## <Mark><C>chk</C></Mark>
## <Item>
## a <C>+</C> sign if at least one program is available for checking
## whether a set of generators is a set of standard generators,
## and
## </Item>
## <Mark><C>prs</C></Mark>
## <Item>
## a <C>+</C> sign if at least one program is available that encodes a
## presentation.
## </Item>
## </List>
## <P/>
## (The list can be printed to the screen or can be fed into a pager,
## see Section <Ref Subsect="subsect:Customizing DisplayAtlasInfo"/>.)
## <P/>
## Called with a list <A>listofnames</A> of strings that are &GAP; names for
## a group from the &ATLAS; of Group Representations,
## <Ref Func="DisplayAtlasInfo"/> prints the overview described above
## but restricted to the groups in this list.
## <P/>
## In addition to or instead of <A>listofnames</A>,
## the string <C>"contents"</C> and a description <A>sources</A> of the
## data may be given about which the overview is formed.
## See below for admissible values of <A>sources</A>.
## <P/>
## Called with a string <A>gapname</A> that is a &GAP; name for a group from
## the &ATLAS; of Group Representations,
## <Ref Func="DisplayAtlasInfo"/> prints an overview of the information
## that is available for this group.
## One line is printed for each faithful representation,
## showing the number of this representation
## (which can be used in calls of <Ref Func="AtlasGenerators"/>),
## and a string of one of the following forms;
## in both cases, <A>id</A> is a (possibly empty) string.
## <P/>
## <List>
## <Mark><C>G <= Sym(<A>n</A><A>id</A>)</C></Mark>
## <Item>
## denotes a permutation representation of degree <A>n</A>,
## for example <C>G <= Sym(40a)</C> and <C>G <= Sym(40b)</C>
## denote two (nonequivalent) representations of degree <M>40</M>.
## </Item>
## <Mark><C>G <= GL(<A>n</A><A>id</A>,<A>descr</A>)</C></Mark>
## <Item>
## denotes a matrix representation of dimension <A>n</A> over a
## coefficient ring described by <A>descr</A>,
## which can be a prime power,
## <C>ℤ</C> (denoting the ring of integers),
## a description of an algebraic extension field,
## <C>ℂ</C> (denoting an unspecified algebraic extension field), or
## <C>ℤ/<A>m</A>ℤ</C> for an integer <A>m</A>
## (denoting the ring of residues mod <A>m</A>);
## for example, <C>G <= GL(2a,4)</C> and <C>G <= GL(2b,4)</C>
## denote two (nonequivalent) representations of dimension <M>2</M> over
## the field with four elements.
## </Item>
## </List>
## <P/>
## After the representations,
## the programs available for <A>gapname</A> are listed.
## <P/>
## The following optional arguments can be used to restrict the overviews.
## <P/>
## <List>
## <Mark><A>std</A></Mark>
## <Item>
## 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 <A>std</A>-th set of standard generators or the
## <M>i</M>-th set of standard generators, for <M>i</M> in <A>std</A>
## (see
## Section <Ref Sect="sect:Standard Generators Used in AtlasRep"/>),
## </Item>
## <Mark><C>"contents"</C> and <A>sources</A></Mark>
## <Item>
## for a string or a list of strings <A>sources</A>,
## restrict the data about which the overview is formed;
## if <A>sources</A> is the string <C>"public"</C> then only non-private
## data
## (see Chapter <Ref Chap="chap:Private Extensions"/>)
## are considered,
## if <A>sources</A> is a string that denotes a private extension in the
## sense of a <A>dirid</A> argument of
## <Ref Func="AtlasOfGroupRepresentationsNotifyPrivateDirectory"/> then
## only the data that belong to this private extension are considered;
## also a list of such strings may be given, then the union of these
## data is considered,
## </Item>
## <Mark><C>Identifier</C> and <A>id</A></Mark>
## <Item>
## restrict to representations with <C>identifier</C> component in the
## list <A>id</A> (note that this component is itself a list, entering
## this list is not admissible),
## or satisfying the function <A>id</A>,
## </Item>
## <Mark><C>IsPermGroup</C> and <K>true</K></Mark>
## <Item>
## restrict to permutation representations,
## </Item>
## <Mark><C>NrMovedPoints</C> and <A>n</A></Mark>
## <Item>
## for a positive integer, a list of positive integers,
## or a property <A>n</A>,
## restrict to permutation representations of degree equal to <A>n</A>,
## or in the list <A>n</A>, or satisfying the function <A>n</A>,
## </Item>
## <Mark><C>NrMovedPoints</C> and the string <C>"minimal"</C></Mark>
## <Item>
## restrict to faithful permutation representations of minimal degree
## (if this information is available),
## </Item>
## <Mark><C>IsTransitive</C> and <K>true</K> or <K>false</K></Mark>
## <Item>
## restrict to transitive or intransitive permutation representations
## (if this information is available),
## </Item>
## <Mark><C>IsPrimitive</C> and <K>true</K> or <K>false</K></Mark>
## <Item>
## restrict to primitive or imprimitive permutation representations
## (if this information is available),
## </Item>
## <Mark><C>Transitivity</C> and <A>n</A></Mark>
## <Item>
## for a nonnegative integer, a list of nonnegative integers,
## or a property <A>n</A>,
## restrict to permutation representations of transitivity equal to
## <A>n</A>, or in the list <A>n</A>, or satisfying the function <A>n</A>
## (if this information is available),
## </Item>
## <Mark><C>RankAction</C> and <A>n</A></Mark>
## <Item>
## for a nonnegative integer, a list of nonnegative integers,
## or a property <A>n</A>,
## restrict to permutation representations of rank equal to
## <A>n</A>, or in the list <A>n</A>, or satisfying the function <A>n</A>
## (if this information is available),
## </Item>
## <Mark><C>IsMatrixGroup</C> and <K>true</K></Mark>
## <Item>
## restrict to matrix representations,
## </Item>
## <Mark><C>Characteristic</C> and <A>p</A></Mark>
## <Item>
## for a prime integer, a list of prime integers, or a property <A>p</A>,
## restrict to matrix representations over fields of characteristic equal
## to <A>p</A>, or in the list <A>p</A>,
## or satisfying the function <A>p</A>
## (representations over residue class rings that are not fields can be
## addressed by entering <K>fail</K> as the value of <A>p</A>),
## </Item>
## <Mark><C>Dimension</C> and <A>n</A></Mark>
## <Item>
## for a positive integer, a list of positive integers,
## or a property <A>n</A>,
## restrict to matrix representations of dimension equal to <A>n</A>,
## or in the list <A>n</A>, or satisfying the function <A>n</A>,
## </Item>
## <Mark><C>Characteristic</C>, <A>p</A>, <C>Dimension</C>,
## and the string <C>"minimal"</C></Mark>
## <Item>
## for a prime integer <A>p</A>,
## restrict to faithful matrix representations over fields
## of characteristic <A>p</A> that have minimal dimension
## (if this information is available),
## </Item>
## <Mark><C>Ring</C> and <A>R</A></Mark>
## <Item>
## for a ring or a property <A>R</A>,
## restrict to matrix representations over this ring
## or satisfying this function
## (note that the representation might be defined over a proper subring
## of <A>R</A>),
## </Item>
## <Mark><C>Ring</C>, <A>R</A>, <C>Dimension</C>,
## and the string <C>"minimal"</C></Mark>
## <Item>
## for a ring <A>R</A>, restrict to faithful matrix representations
## over this ring that have minimal dimension
## (if this information is available),
## </Item>
## <Mark><C>Character</C> and <A>chi</A></Mark>
## <Item>
## for a class function or a list of class functions <A>chi</A>,
## restrict to matrix representations with these characters
## (note that the underlying characteristic of the class function,
## see Section <Ref Sect="UnderlyingCharacteristic" BookName="ref"/>,
## determines the characteristic of the matrices),
## and
## </Item>
## <Mark><C>IsStraightLineProgram</C> and <K>true</K></Mark>
## <Item>
## restrict to straight line programs,
## straight line decisions
## (see Section <Ref Sect="sect:Straight Line Decisions"/>),
## and black box programs
## (see Section <Ref Sect="sect:Black Box Programs"/>).
## </Item>
## </List>
## <P/>
## Note that the above conditions refer only to the information that is
## available without accessing the representations.
## For example, if it is not stored in the table of contents whether a
## permutation representation is primitive then this representation does not
## match an <C>IsPrimitive</C> condition in <Ref Func="DisplayAtlasInfo"/>.
## <P/>
## If <Q>minimality</Q> information is requested and no available
## representation matches this condition then either no minimal
## representation is available or the information about the minimality
## is missing.
## See <Ref Func="MinimalRepresentationInfo"/> for checking whether the
## minimality information is available for the group in question.
## Note that in the cases where the string <C>"minimal"</C> occurs as an
## argument, <Ref Func="MinimalRepresentationInfo"/> is called with third
## argument <C>"lookup"</C>;
## this is because the stored information was precomputed just for
## the groups in the &ATLAS; of Group Representations,
## so trying to compute non-stored minimality information (using other
## available databases) will hardly be successful.
## <P/>
## The representations are ordered as follows.
## Permutation representations come first (ordered according to their
## degrees),
## followed by matrix representations over finite fields
## (ordered first according to the field size 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).
## <P/>
## The maximal subgroups are ordered according to decreasing group order.
## For an extension <M>G.p</M> of a simple group <M>G</M> by an outer
## automorphism of prime order <M>p</M>,
## this means that <M>G</M> is the first maximal subgroup
## and then come the extensions of the maximal subgroups of <M>G</M> and the
## novelties;
## so the <M>n</M>-th maximal subgroup of <M>G</M> and the <M>n</M>-th
## maximal subgroup of <M>G.p</M> are in general not related.
## (This coincides with the numbering used for the
## <Ref Func="Maxes" BookName="ctbllib"/> attribute for character tables.)
## <P/>
## <Example><![CDATA[
## gap> DisplayAtlasInfo( [ "M11", "A5" ] );
## group | # | maxes | cl | cyc | out | fnd | chk | prs
## ------+----+-------+----+-----+-----+-----+-----+----
## M11 | 42 | 5 | + | + | | + | + | +
## A5 | 18 | 3 | | | | | + | +
## ]]></Example>
## <P/>
## The above output means that the &ATLAS; of Group Representations contains
## <M>42</M> representations of the Mathieu group <M>M_{11}</M>,
## 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,
## contains no straight line program for applying outer automorphisms
## (well, in fact <M>M_{11}</M> admits no nontrivial outer automorphism),
## and contains straight line decisions that check a set of generators
## or a set of group elements for being a set of standard generators.
## Analogously,
## <M>18</M> representations of the alternating group <M>A_5</M> 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;
## straight line decisions for checking the standardization of generators
## or group elements are available.
## <P/>
## <Example><![CDATA[
## gap> DisplayAtlasInfo( "A5", IsPermGroup, true );
## Representations for G = A5: (all refer to std. generators 1)
## ---------------------------
## 1: G <= Sym(5) 3-trans., on cosets of A4 (1st max.)
## 2: G <= Sym(6) 2-trans., on cosets of D10 (2nd max.)
## 3: G <= Sym(10) rank 3, on cosets of S3 (3rd max.)
## gap> DisplayAtlasInfo( "A5", NrMovedPoints, [ 4 .. 9 ] );
## Representations for G = A5: (all refer to std. generators 1)
## ---------------------------
## 1: G <= Sym(5) 3-trans., on cosets of A4 (1st max.)
## 2: G <= Sym(6) 2-trans., on cosets of D10 (2nd max.)
## ]]></Example>
## <P/>
## The first three representations stored for <M>A_5</M> are
## (in fact primitive) permutation representations.
## <P/>
## <Example><![CDATA[
## 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)]))
## ]]></Example>
## <P/>
## The representations with number between <M>4</M> and <M>13</M> are
## (in fact irreducible) matrix representations over various finite fields,
## those with numbers <M>14</M> to <M>16</M> are integral matrix
## representations,
## and the last two are matrix representations over the field generated by
## <M>\sqrt{{5}}</M> over the rational number field.
## <P/>
## <Example><![CDATA[
## gap> DisplayAtlasInfo( "A5", Identifier, "a" );
## Representations for G = A5: (all refer to std. generators 1)
## ---------------------------
## 4: G <= GL(4a,2)
## 8: G <= GL(2a,4)
## 12: G <= GL(3a,9)
## 17: G <= GL(3a,Field([Sqrt(5)]))
## ]]></Example>
## <P/>
## Each of the representations with the numbers <M>4, 8, 12</M>,
## and <M>17</M> is labeled with the distinguishing letter <C>a</C>.
## <P/>
## <Example><![CDATA[
## gap> DisplayAtlasInfo( "A5", NrMovedPoints, IsPrimeInt );
## Representations for G = A5: (all refer to std. generators 1)
## ---------------------------
## 1: G <= Sym(5) 3-trans., on cosets of A4 (1st max.)
## gap> DisplayAtlasInfo( "A5", Characteristic, IsOddInt );
## Representations for G = A5: (all refer to std. generators 1)
## ---------------------------
## 6: G <= GL(4,3)
## 7: G <= GL(6,3)
## 10: G <= GL(3,5)
## 11: G <= GL(5,5)
## 12: G <= GL(3a,9)
## 13: G <= GL(3b,9)
## gap> DisplayAtlasInfo( "A5", Dimension, IsPrimeInt );
## 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)
## 11: G <= GL(5,5)
## 12: G <= GL(3a,9)
## 13: G <= GL(3b,9)
## 15: G <= GL(5,Z)
## 17: G <= GL(3a,Field([Sqrt(5)]))
## 18: G <= GL(3b,Field([Sqrt(5)]))
## gap> DisplayAtlasInfo( "A5", Ring, IsFinite and IsPrimeField );
## Representations for G = A5: (all refer to std. generators 1)
## ---------------------------
## 4: G <= GL(4a,2)
## 5: G <= GL(4b,2)
## 6: G <= GL(4,3)
## 7: G <= GL(6,3)
## 10: G <= GL(3,5)
## 11: G <= GL(5,5)
## ]]></Example>
## <P/>
## The above examples show how the output can be restricted using a property
## (a unary function that returns either <K>true</K> or <K>false</K>)
## that follows <Ref Func="NrMovedPoints" BookName="ref"/>,
## <Ref Func="Characteristic" BookName="ref"/>,
## <Ref Func="Dimension" BookName="ref"/>,
## or <Ref Func="Ring" BookName="ref"/>
## in the argument list of <Ref Func="DisplayAtlasInfo"/>.
## <P/>
## <Example><![CDATA[
## gap> DisplayAtlasInfo( "A5", IsStraightLineProgram, true );
## Programs for G = A5: (all refer to std. generators 1)
## --------------------
## presentation
## std. gen. checker
## maxes (all 3):
## 1: A4
## 2: D10
## 3: S3
## ]]></Example>
## <P/>
## Straight line programs are available for computing generators of
## representatives of the three classes of maximal subgroups of <M>A_5</M>,
## and a straight line decision for checking whether given generators are
## in fact standard generators is available as well as a presentation
## in terms of standard generators,
## see <Ref Func="AtlasProgram"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "DisplayAtlasInfo" );
#############################################################################
##
#F AtlasGenerators( <gapname>, <repnr>[, <maxnr>] )
#F AtlasGenerators( <identifier> )
##
## <#GAPDoc Label="AtlasGenerators">
## <ManSection>
## <Func Name="AtlasGenerators" Arg='gapname, repnr[, maxnr]'/>
## <Func Name="AtlasGenerators" Arg='identifier' Label="for an identifier"/>
##
## <Returns>
## a record containing generators for a representation, or <K>fail</K>.
## </Returns>
## <Description>
## In the first form, <A>gapname</A> must be a string denoting a &GAP; name
## (see
## Section <Ref Sect="sect:Group Names Used in the AtlasRep Package"/>)
## of a group, and <A>repnr</A> a positive integer.
## If the &ATLAS; of Group Representations contains at least <A>repnr</A>
## representations for the group with &GAP; name <A>gapname</A> then
## <Ref Func="AtlasGenerators"/>,
## when called with <A>gapname</A> and <A>repnr</A>,
## returns an immutable record describing the <A>repnr</A>-th
## representation;
## otherwise <K>fail</K> is returned.
## If a third argument <A>maxnr</A>, a positive integer,
## is given then an immutable record describing the restriction of the
## <A>repnr</A>-th representation to the <A>maxnr</A>-th maximal subgroup is
## returned.
## <P/>
## The result record has at least the following components.
## <P/>
## <List>
## <Mark><C>generators</C></Mark>
## <Item>
## a list of generators for the group,
## </Item>
## <Mark><C>groupname</C></Mark>
## <Item>
## the &GAP; name of the group (see
## Section <Ref Sect="sect:Group Names Used in the AtlasRep Package"/>),
## </Item>
## <Mark><C>identifier</C></Mark>
## <Item>
## a &GAP; object (a list of filenames plus additional information)
## that uniquely determines the representation;
## the value can be used as <A>identifier</A> argument of
## <Ref Func="AtlasGenerators"/>.
## </Item>
## <Mark><C>repnr</C></Mark>
## <Item>
## the number of the representation in the current session,
## equal to the argument <A>repnr</A> if this is given.
## </Item>
## <Mark><C>standardization</C></Mark>
## <Item>
## the positive integer denoting the underlying standard generators,
## </Item>
## </List>
## <P/>
## Additionally, the group order may be stored in the component <C>size</C>,
## and describing components may be available that depend on the data type
## of the representation:
## For permutation representations, these are <C>p</C> for the number of
## moved points, <C>id</C> for the distinguishing string as described for
## <Ref Func="DisplayAtlasInfo"/>, and information about primitivity,
## point stabilizers etc. if available;
## for matrix representations, these are <C>dim</C> for the dimension of the
## matrices, <C>ring</C> (if known) for the ring generated by the matrix
## entries, <C>id</C> for the distinguishing string, and information about
## the character if available.
## <P/>
## It should be noted that the number <A>repnr</A> refers to the number
## shown by <Ref Func="DisplayAtlasInfo"/> <E>in the current session</E>;
## it may be that after the addition of new representations,
## <A>repnr</A> refers to another representation.
## <P/>
## The alternative form of <Ref Func="AtlasGenerators"/>,
## with only argument <A>identifier</A>,
## can be used to fetch the result record with <C>identifier</C> value equal
## to <A>identifier</A>.
## The purpose of this variant is to access the <E>same</E> representation
## also in <E>different</E> &GAP; sessions.
## <P/>
## <Example><![CDATA[
## gap> gens1:= AtlasGenerators( "A5", 1 );
## rec( generators := [ (1,2)(3,4), (1,3,5) ], groupname := "A5",
## id := "",
## identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ],
## isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,
## repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4",
## standardization := 1, transitivity := 3, type := "perm" )
## gap> gens8:= AtlasGenerators( "A5", 8 );
## rec( dim := 2,
## 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 ] ] ], groupname := "A5",
## id := "a",
## identifier := [ "A5", [ "A5G1-f4r2aB0.m1", "A5G1-f4r2aB0.m2" ], 1,
## 4 ], repname := "A5G1-f4r2aB0", repnr := 8, ring := GF(2^2),
## size := 60, standardization := 1, type := "matff" )
## gap> gens17:= AtlasGenerators( "A5", 17 );
## rec( dim := 3,
## 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 ] ] ],
## groupname := "A5", id := "a",
## identifier := [ "A5", "A5G1-Ar3aB0.g", 1, 3 ],
## repname := "A5G1-Ar3aB0", repnr := 17, ring := NF(5,[ 1, 4 ]),
## size := 60, standardization := 1, type := "matalg" )
## ]]></Example>
## <P/>
## Each of the above pairs of elements generates a group isomorphic to
## <M>A_5</M>.
## <P/>
## <Example><![CDATA[
## gap> gens1max2:= AtlasGenerators( "A5", 1, 2 );
## rec( generators := [ (1,2)(3,4), (2,3)(4,5) ], groupname := "D10",
## identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5, 2 ],
## repnr := 1, size := 10, standardization := 1 )
## 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
## ]]></Example>
## <P/>
## The elements stored in <C>gens1max2.generators</C> describe the
## restriction of the first representation of <M>A_5</M> to a group in the
## second class of maximal subgroups of <M>A_5</M> according to the list in
## the &ATLAS; of Finite Groups <Cite Key="CCN85"/>;
## this subgroup is isomorphic to the dihedral group <M>D_{10}</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "AtlasGenerators" );
#############################################################################
##
#F AtlasProgramInfo( <gapname>[, <std>][, "maxes"], <maxnr> )
#F AtlasProgramInfo( <gapname>[, <std>], "classes" )
#F AtlasProgramInfo( <gapname>[, <std>], "cyclic" )
#F AtlasProgramInfo( <gapname>[, <std>], "automorphism", <autname> )
#F AtlasProgramInfo( <gapname>[, <std>], "check" )
#F AtlasProgramInfo( <gapname>[, <std>], "pres" )
#F AtlasProgramInfo( <gapname>[, <std>], "find" )
#F AtlasProgramInfo( <gapname>, <std>, "restandardize", <std2> )
#F AtlasProgramInfo( <gapname>[, <std>], "other", <descr> )
##
## <#GAPDoc Label="AtlasProgramInfo">
## <ManSection>
## <Func Name="AtlasProgramInfo"
## Arg='gapname[, std][, "contents", sources][, ...]'/>
##
## <Returns>
## a record describing a program, or <K>fail</K>.
## </Returns>
## <Description>
## <Ref Func="AtlasProgramInfo"/> takes the same arguments as
## <Ref Func="AtlasProgram"/>, and returns a similar result.
## The only difference is that the records returned by
## <Ref Func="AtlasProgramInfo"/> have no components <C>program</C> and
## <C>outputs</C>.
## The idea is that one can use <Ref Func="AtlasProgramInfo"/> for
## testing whether the program in question is available at all,
## but without transferring it from a remote server.
## The <C>identifier</C> component of the result of
## <Ref Func="AtlasProgramInfo"/> can then be used to fetch the program
## with <Ref Func="AtlasProgram"/>.
##
## <Example><![CDATA[
## gap> AtlasProgramInfo( "J1", "cyclic" );
## rec( groupname := "J1", identifier := [ "J1", "J1G1-cycW1", 1 ],
## standardization := 1 )
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "AtlasProgramInfo" );
#############################################################################
##
#F AtlasProgram( <gapname>[, <std>][, "maxes"], <maxnr> )
#F AtlasProgram( <gapname>[, <std>], "classes" )
#F AtlasProgram( <gapname>[, <std>], "cyclic" )
#F AtlasProgram( <gapname>[, <std>], "automorphism", <autname> )
#F AtlasProgram( <gapname>[, <std>], "check" )
#F AtlasProgram( <gapname>[, <std>], "presentation" )
#F AtlasProgram( <gapname>[, <std>], "find" )
#F AtlasProgram( <gapname>, <std>, "restandardize", <std2> )
#F AtlasProgram( <gapname>[, <std>], "other", <descr> )
#F AtlasProgram( <identifier> )
##
## <#GAPDoc Label="AtlasProgram">
## <ManSection>
## <Func Name="AtlasProgram" Arg='gapname[, std], ...'/>
## <Func Name="AtlasProgram" Arg='identifier' Label="for an identifier"/>
##
## <Returns>
## a record containing a program, or <K>fail</K>.
## </Returns>
## <Description>
## In the first form, <A>gapname</A> must be a string denoting a &GAP; name
## (see Section
## <Ref Sect="sect:Group Names Used in the AtlasRep Package"/>)
## of a group <M>G</M>, say.
## If the &ATLAS; of Group Representations contains a straight line program
## (see Section <Ref Sect="Straight Line Programs" BookName="ref"/>)
## or straight line decision
## (see Section <Ref Sect="sect:Straight Line Decisions"/>)
## or black box program
## (see Section <Ref Sect="sect:Black Box Programs"/>)
## as described by the remaining arguments (see below) then
## <Ref Func="AtlasProgram"/> returns an immutable record
## containing this program.
## Otherwise <K>fail</K> is returned.
## <P/>
## If the optional argument <A>std</A> is given, only those straight line
## programs/decisions are considered
## that take generators from the <A>std</A>-th set
## of standard generators of <M>G</M> as input,
## see Section <Ref Sect="sect:Standard Generators Used in AtlasRep"/>.
## <P/>
## The result record has the following components.
## <P/>
## <List>
## <Mark><C>program</C></Mark>
## <Item>
## the required straight line program/decision, or black box program,
## </Item>
## <Mark><C>standardization</C></Mark>
## <Item>
## the positive integer denoting the underlying standard generators of
## <M>G</M>,
## </Item>
## <Mark><C>identifier</C></Mark>
## <Item>
## a &GAP; object (a list of filenames plus additional information)
## that uniquely determines the program;
## the value can be used as <A>identifier</A> argument of
## <Ref Func="AtlasProgram"/> (see below).
## </Item>
## </List>
## <P/>
## In the first form, the last arguments must be as follows.
## <P/>
## <List>
## <Mark>(the string <C>"maxes"</C> and) a positive integer <A>maxnr</A>
## </Mark>
## <Item>
## the required program computes generators of the <A>maxnr</A>-th
## maximal subgroup of the group with &GAP; name <A>gapname</A>.
## <Index Subkey="for maximal subgroups">straight line program</Index>
## <Index>maximal subgroups</Index>
## <P/>
## In this case, the result record of <Ref Func="AtlasProgram"/> also
## may contain a component <C>size</C>,
## whose value is the order of the maximal subgroup in question.
## </Item>
## <Mark>one of the strings <C>"classes"</C> or <C>"cyclic"</C></Mark>
## <Item>
## the required program computes representatives of conjugacy classes
## of elements or representatives of generators of maximally cyclic
## subgroups of <M>G</M>, respectively.
## <Index Subkey="for class representatives">straight line program</Index>
## <Index>class representatives</Index>
## <Index Subkey="for representatives of cyclic subgroups">
## straight line program</Index>
## <Index>cyclic subgroups</Index>
## <Index>maximally cyclic subgroups</Index>
## <P/>
## See <Cite Key="BSW01"/> and <Cite Key="SWW00"/>
## for the background concerning these straight line programs.
## In these cases, the result record of <Ref Func="AtlasProgram"/>
## also contains a component <C>outputs</C>,
## whose value is a list of class names of the outputs,
## as described in
## Section <Ref Sect="sect:Class Names Used in the AtlasRep Package"/>.
## </Item>
## <Mark>the strings <C>"automorphism"</C> and <A>autname</A></Mark>
## <Item>
## <Index Subkey="for outer automorphisms">straight line program</Index>
## <Index>automorphisms</Index>
## the required program computes images of standard generators under
## the outer automorphism of <M>G</M> that is given by this string.
## <P/>
## Note that a value <C>"2"</C> of <A>autname</A> means that the square of
## the automorphism is an inner automorphism of <M>G</M> (not necessarily
## the identity mapping) but the automorphism itself is not.
## </Item>
## <Mark>the string <C>"check"</C></Mark>
## <Item>
## <Index Subkey="for checking standard generators">straight line decision
## </Index>
## the required result is a straight line decision that
## takes a list of generators for <M>G</M>
## and returns <K>true</K> if these generators are standard generators of
## <M>G</M> w.r.t. the standardization <A>std</A>,
## and <K>false</K> otherwise.
## </Item>
## <Mark>the string <C>"presentation"</C></Mark>
## <Item>
## <Index Subkey="encoding a presentation">straight line decision
## </Index>
## the required result is a straight line decision that
## takes a list of group elements
## and returns <K>true</K> if these elements are standard generators of
## <M>G</M> w.r.t. the standardization <A>std</A>,
## and <K>false</K> otherwise.
## <P/>
## See <Ref Func="StraightLineProgramFromStraightLineDecision"/> for an
## example how to derive defining relators for <M>G</M> in terms of the
## standard generators from such a straight line decision.
## </Item>
## <Mark>the string <C>"find"</C></Mark>
## <Item>
## <Index Subkey="for finding standard generators">black box program
## </Index>
## the required result is a black box program that takes <M>G</M>
## and returns a list of standard generators of <M>G</M>,
## w.r.t. the standardization <A>std</A>.
## </Item>
## <Mark>the string <C>"restandardize"</C> and an integer <A>std2</A></Mark>
## <Item>
## <Index Subkey="for restandardizing">straight line program</Index>
## the required result is a straight line program that computes
## standard generators of <M>G</M> w.r.t. the <A>std2</A>-th set
## of standard generators of <M>G</M>;
## in this case, the argument <A>std</A> must be given.
## </Item>
## <Mark>the strings <C>"other"</C> and <A>descr</A></Mark>
## <Item>
## <Index Subkey="free format">straight line program</Index>
## the required program is described by <A>descr</A>.
## </Item>
## </List>
## <P/>
## The second form of <Ref Func="AtlasProgram"/>,
## with only argument the list <A>identifier</A>,
## can be used to fetch the result record with <C>identifier</C> value equal
## to <A>identifier</A>.
## <Example><![CDATA[
## gap> prog:= AtlasProgram( "A5", 2 );
## rec( groupname := "A5", identifier := [ "A5", "A5G1-max2W1", 1 ],
## program := <straight line program>, size := 10,
## standardization := 1, subgroupname := "D10" )
## gap> StringOfResultOfStraightLineProgram( prog.program, [ "a", "b" ] );
## "[ a, bbab ]"
## gap> gens1:= AtlasGenerators( "A5", 1 );
## rec( generators := [ (1,2)(3,4), (1,3,5) ], groupname := "A5",
## id := "",
## identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ],
## isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,
## repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4",
## standardization := 1, transitivity := 3, type := "perm" )
## gap> maxgens:= ResultOfStraightLineProgram( prog.program, gens1.generators );
## [ (1,2)(3,4), (2,3)(4,5) ]
## gap> maxgens = gens1max2.generators;
## true
## ]]></Example>
## <P/>
## The above example shows that for restricting representations given by
## standard generators to a maximal subgroup of <M>A_5</M>,
## we can also fetch and apply the appropriate straight line program.
## Such a program
## (see <Ref Sect="Straight Line Programs" BookName="ref"/>)
## takes standard generators of a group --in this example <M>A_5</M>--
## as its input, and returns a list of elements in this group
## --in this example generators of the <M>D_{10}</M> subgroup we had met
## above--
## which are computed essentially by evaluating structured words in terms of
## the standard generators.
## <P/>
## <Example><![CDATA[
## gap> prog:= AtlasProgram( "J1", "cyclic" );
## rec( groupname := "J1", identifier := [ "J1", "J1G1-cycW1", 1 ],
## outputs := [ "6A", "7A", "10B", "11A", "15B", "19A" ],
## program := <straight line program>, standardization := 1 )
## gap> gens:= GeneratorsOfGroup( FreeGroup( "x", "y" ) );;
## gap> ResultOfStraightLineProgram( prog.program, gens );
## [ (x*y)^2*((y*x)^2*y^2*x)^2*y^2, x*y, (x*(y*x*y)^2)^2*y,
## (x*y*x*(y*x*y)^3*x*y^2)^2*x*y*x*(y*x*y)^2*y, x*y*x*(y*x*y)^2*y,
## (x*y)^2*y ]
## ]]></Example>
## <P/>
## 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.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "AtlasProgram" );
#############################################################################
##
#F OneAtlasGeneratingSetInfo( [<gapname>][, <std>] )
#F OneAtlasGeneratingSetInfo( [<gapname>][, <std>], IsPermGroup[, true] )
#F OneAtlasGeneratingSetInfo( [<gapname>][, <std>], NrMovedPoints, <n> )
#F OneAtlasGeneratingSetInfo( [<gapname>][, <std>], IsMatrixGroup[, true] )
#F OneAtlasGeneratingSetInfo( [<gapname>][, <std>][, Characteristic, <p>]
#F [, Dimension, <m>] )
#F OneAtlasGeneratingSetInfo( [<gapname>][, <std>][, Ring, <R>]
#F [, Dimension, <m>] )
#F OneAtlasGeneratingSetInfo( [<gapname>,][ <std>,] Position, <n> )
##
## <#GAPDoc Label="OneAtlasGeneratingSetInfo">
## <ManSection>
## <Func Name="OneAtlasGeneratingSetInfo" Arg='[gapname][, std][, ...]'/>
##
## <Returns>
## a record describing a representation that satisfies the conditions,
## or <K>fail</K>.
## </Returns>
## <Description>
## Let <A>gapname</A> be a string denoting a &GAP; name (see Section
## <Ref Sect="sect:Group Names Used in the AtlasRep Package"/>)
## of a group <M>G</M>, say.
## If the &ATLAS; of Group Representations contains at least one
## representation for <M>G</M> with the required properties
## then <Ref Func="OneAtlasGeneratingSetInfo"/> returns a record <A>r</A>
## whose components are the same as those of the records returned by
## <Ref Func="AtlasGenerators"/>,
## except that the component <C>generators</C> is not contained;
## the component <C>identifier</C> of <A>r</A> can be used as input for
## <Ref Func="AtlasGenerators"/> in order to fetch the generators.
## If no representation satisfying the given conditions is available
## then <K>fail</K> is returned.
## <P/>
## If the argument <A>std</A> 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
## Section <Ref Sect="sect:Standard Generators Used in AtlasRep"/>).
## <P/>
## The argument <A>gapname</A> can be missing (then all available groups are
## considered), or a list of group names can be given instead.
## <P/>
## Further restrictions can be entered as arguments, with the same meaning
## as described for <Ref Func="DisplayAtlasInfo"/>.
## The result of <Ref Func="OneAtlasGeneratingSetInfo"/> describes the first
## generating set for <M>G</M> that matches the restrictions,
## in the ordering shown by <Ref Func="DisplayAtlasInfo"/>.
## <P/>
## Note that even in the case that the user parameter <Q>remote</Q>
## has the value <K>true</K>
## (see Section <Ref Subsect="subsect:Local or remote access"/>),
## <Ref Func="OneAtlasGeneratingSetInfo"/> does <E>not</E> attempt
## to <E>transfer</E> remote data files,
## just the table of contents is evaluated.
## So this function (as well as <Ref Func="AllAtlasGeneratingSetInfos"/>)
## can be used to check for the availability of certain representations,
## and afterwards one can call <Ref Func="AtlasGenerators"/> for those
## representations one wants to work with.
## <P/>
## In the following example, we try to access information about
## permutation representations for the alternating group <M>A_5</M>.
## <P/>
## <Example><![CDATA[
## gap> info:= OneAtlasGeneratingSetInfo( "A5" );
## rec( groupname := "A5", id := "",
## identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ],
## isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,
## repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4",
## standardization := 1, transitivity := 3, type := "perm" )
## gap> gens:= AtlasGenerators( info.identifier );
## rec( generators := [ (1,2)(3,4), (1,3,5) ], groupname := "A5",
## id := "",
## identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ],
## isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,
## repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4",
## standardization := 1, transitivity := 3, type := "perm" )
## gap> info = OneAtlasGeneratingSetInfo( "A5", IsPermGroup, true );
## true
## gap> info = OneAtlasGeneratingSetInfo( "A5", NrMovedPoints, "minimal" );
## true
## gap> info = OneAtlasGeneratingSetInfo( "A5", NrMovedPoints, [ 1 .. 10 ] );
## true
## gap> OneAtlasGeneratingSetInfo( "A5", NrMovedPoints, 20 );
## fail
## ]]></Example>
## <P/>
## Note that a permutation representation of degree <M>20</M> could be
## obtained by taking twice the primitive representation on <M>10</M> points;
## however, the &ATLAS; of Group Representations does not store this
## imprimitive representation (cf.
## Section <Ref Sect="sect:Accessing vs. Constructing Representations"/>).
## <P/>
## We continue this example a little.
## Next we access matrix representations of <M>A_5</M>.
## <P/>
## <Example><![CDATA[
## gap> info:= OneAtlasGeneratingSetInfo( "A5", IsMatrixGroup, true );
## rec( dim := 4, groupname := "A5", id := "a",
## identifier := [ "A5", [ "A5G1-f2r4aB0.m1", "A5G1-f2r4aB0.m2" ], 1,
## 2 ], repname := "A5G1-f2r4aB0", repnr := 4, ring := GF(2),
## size := 60, standardization := 1, type := "matff" )
## gap> gens:= AtlasGenerators( info.identifier );
## rec( dim := 4,
## generators := [ <an immutable 4x4 matrix over GF2>,
## <an immutable 4x4 matrix over GF2> ], groupname := "A5",
## id := "a",
## identifier := [ "A5", [ "A5G1-f2r4aB0.m1", "A5G1-f2r4aB0.m2" ], 1,
## 2 ], repname := "A5G1-f2r4aB0", repnr := 4, ring := GF(2),
## size := 60, standardization := 1, type := "matff" )
## gap> info = OneAtlasGeneratingSetInfo( "A5", Dimension, 4 );
## true
## gap> info = OneAtlasGeneratingSetInfo( "A5", Characteristic, 2 );
## true
## gap> info = OneAtlasGeneratingSetInfo( "A5", Ring, GF(2) );
## true
## gap> OneAtlasGeneratingSetInfo( "A5", Characteristic, [2,5], Dimension, 2 );
## rec( dim := 2, groupname := "A5", id := "a",
## identifier := [ "A5", [ "A5G1-f4r2aB0.m1", "A5G1-f4r2aB0.m2" ], 1,
## 4 ], repname := "A5G1-f4r2aB0", repnr := 8, ring := GF(2^2),
## size := 60, standardization := 1, type := "matff" )
## gap> OneAtlasGeneratingSetInfo( "A5", Characteristic, [2,5], Dimension, 1 );
## fail
## gap> info:= OneAtlasGeneratingSetInfo( "A5", Characteristic, 0, Dimension, 4 );
## rec( dim := 4, groupname := "A5", id := "",
## identifier := [ "A5", "A5G1-Zr4B0.g", 1, 4 ],
## repname := "A5G1-Zr4B0", repnr := 14, ring := Integers, size := 60,
## standardization := 1, type := "matint" )
## gap> gens:= AtlasGenerators( info.identifier );
## rec( dim := 4,
## 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 ] ] ], groupname := "A5", id := "",
## identifier := [ "A5", "A5G1-Zr4B0.g", 1, 4 ],
## repname := "A5G1-Zr4B0", repnr := 14, ring := Integers, size := 60,
## standardization := 1, type := "matint" )
## gap> info = OneAtlasGeneratingSetInfo( "A5", Ring, Integers );
## true
## gap> info = OneAtlasGeneratingSetInfo( "A5", Ring, CF(37) );
## true
## gap> OneAtlasGeneratingSetInfo( "A5", Ring, Integers mod 77 );
## fail
## gap> info:= OneAtlasGeneratingSetInfo( "A5", Ring, CF(5), Dimension, 3 );
## rec( dim := 3, groupname := "A5", id := "a",
## identifier := [ "A5", "A5G1-Ar3aB0.g", 1, 3 ],
## repname := "A5G1-Ar3aB0", repnr := 17, ring := NF(5,[ 1, 4 ]),
## size := 60, standardization := 1, type := "matalg" )
## gap> gens:= AtlasGenerators( info.identifier );
## rec( dim := 3,
## 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 ] ] ],
## groupname := "A5", id := "a",
## identifier := [ "A5", "A5G1-Ar3aB0.g", 1, 3 ],
## repname := "A5G1-Ar3aB0", repnr := 17, ring := NF(5,[ 1, 4 ]),
## size := 60, standardization := 1, type := "matalg" )
## gap> OneAtlasGeneratingSetInfo( "A5", Ring, GF(17) );
## fail
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "OneAtlasGeneratingSetInfo" );
#############################################################################
##
#F AllAtlasGeneratingSetInfos( [<gapname>][, <std>] )
#F AllAtlasGeneratingSetInfos( [<gapname>][, <std>], IsPermGroup[, true] )
#F AllAtlasGeneratingSetInfos( [<gapname>][, <std>], NrMovedPoints, <n> )
#F AllAtlasGeneratingSetInfos( [<gapname>][, <std>], IsMatrixGroup[, true] )
#F AllAtlasGeneratingSetInfos( [<gapname>][, <std>][, Characteristic, <p>]
#F [, Dimension, <m>] )
#F AllAtlasGeneratingSetInfos( [<gapname>][, <std>][, Ring, <R>]
#F [, Dimension, <m>] )
##
## <#GAPDoc Label="AllAtlasGeneratingSetInfos">
## <ManSection>
## <Func Name="AllAtlasGeneratingSetInfos" Arg='[gapname][, std][, ...]'/>
##
## <Returns>
## the list of all records describing representations that satisfy
## the conditions.
## </Returns>
## <Description>
## <Ref Func="AllAtlasGeneratingSetInfos"/> is similar to
## <Ref Func="OneAtlasGeneratingSetInfo"/>.
## The difference is that the list of <E>all</E> records describing
## the available representations with the given properties is returned
## instead of just one such component.
## In particular an empty list is returned if no such representation is
## available.
## <P/>
## <Example><![CDATA[
## gap> AllAtlasGeneratingSetInfos( "A5", IsPermGroup, true );
## [ rec( groupname := "A5", id := "",
## identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ]
## , isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,
## repname := "A5G1-p5B0", repnr := 1, size := 60,
## stabilizer := "A4", standardization := 1, transitivity := 3,
## type := "perm" ),
## rec( groupname := "A5", id := "",
## identifier := [ "A5", [ "A5G1-p6B0.m1", "A5G1-p6B0.m2" ], 1, 6 ]
## , isPrimitive := true, maxnr := 2, p := 6, rankAction := 2,
## repname := "A5G1-p6B0", repnr := 2, size := 60,
## stabilizer := "D10", standardization := 1, transitivity := 2,
## type := "perm" ),
## rec( groupname := "A5", id := "",
## identifier := [ "A5", [ "A5G1-p10B0.m1", "A5G1-p10B0.m2" ], 1,
## 10 ], isPrimitive := true, maxnr := 3, p := 10,
## rankAction := 3, repname := "A5G1-p10B0", repnr := 3,
## size := 60, stabilizer := "S3", standardization := 1,
## transitivity := 1, type := "perm" ) ]
## ]]></Example>
## <P/>
## 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 <E>store</E> such a
## representation
## (cf. Section <Ref Sect="sect:Accessing vs. Constructing Representations"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "AllAtlasGeneratingSetInfos" );
#############################################################################
##
#A AtlasRepInfoRecord( <G> )
##
## <#GAPDoc Label="AtlasRepInfoRecord">
## <ManSection>
## <Attr Name="AtlasRepInfoRecord" Arg='G'/>
## <Returns>
## the record stored in the group <A>G</A> when this was constructed
## with <Ref Func="AtlasGroup" Label="for various arguments"/>.
## </Returns>
## <Description>
## For a group <A>G</A> that has been constructed with
## <Ref Func="AtlasGroup" Label="for various arguments"/>,
## the value of this attribute is the info record that describes <A>G</A>,
## in the sense that this record was the first argument of the call to
## <Ref Func="AtlasGroup" Label="for various arguments"/>, or it is the
## result of the call to <Ref Func="OneAtlasGeneratingSetInfo"/> with the
## conditions that were listed in the call to
## <Ref Func="AtlasGroup" Label="for various arguments"/>.
## <P/>
## <Example><![CDATA[
## gap> AtlasRepInfoRecord( AtlasGroup( "A5" ) );
## rec( groupname := "A5", id := "",
## identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ],
## isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,
## repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4",
## standardization := 1, transitivity := 3, type := "perm" )
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "AtlasRepInfoRecord", IsGroup );
#############################################################################
##
#F AtlasGroup( [<gapname>[, <std>]] )
#F AtlasGroup( [<gapname>[, <std>]], IsPermGroup[, true] )
#F AtlasGroup( [<gapname>[, <std>]], NrMovedPoints, <n> )
#F AtlasGroup( [<gapname>[, <std>]], IsMatrixGroup[, true] )
#F AtlasGroup( [<gapname>[, <std>]][, Characteristic, <p>]
#F [, Dimension, <m>] )
#F AtlasGroup( [<gapname>[, <std>]][, Ring, <R>][, Dimension, <m>] )
#F AtlasGroup( <identifier> )
##
## <#GAPDoc Label="AtlasGroup">
## <ManSection>
## <Heading>AtlasGroup</Heading>
## <Func Name="AtlasGroup" Arg='[gapname[, std]][, ...]'
## Label="for various arguments"/>
## <Func Name="AtlasGroup" Arg='identifier'
## Label="for an identifier record"/>
##
## <Returns>
## a group that satisfies the conditions, or <K>fail</K>.
## </Returns>
## <Description>
## <Ref Func="AtlasGroup" Label="for various arguments"/> takes the same
## arguments as <Ref Func="OneAtlasGeneratingSetInfo"/>,
## and returns the group generated by the <C>generators</C> component
## of the record that is returned by <Ref Func="OneAtlasGeneratingSetInfo"/>
## with these arguments;
## if <Ref Func="OneAtlasGeneratingSetInfo"/> returns <K>fail</K> then also
## <Ref Func="AtlasGroup" Label="for various arguments"/> returns
## <K>fail</K>.
## <P/>
## <Example><![CDATA[
## gap> g:= AtlasGroup( "A5" );
## Group([ (1,2)(3,4), (1,3,5) ])
## ]]></Example>
## <P/>
## Alternatively, it is possible to enter exactly one argument,
## a record <A>identifier</A> as returned by
## <Ref Func="OneAtlasGeneratingSetInfo"/> or
## <Ref Func="AllAtlasGeneratingSetInfos"/>,
## or the <C>identifier</C> component of such a record.
## <P/>
## <Example><![CDATA[
## gap> info:= OneAtlasGeneratingSetInfo( "A5" );
## rec( groupname := "A5", id := "",
## identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ],
## isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,
## repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4",
## standardization := 1, transitivity := 3, type := "perm" )
## gap> AtlasGroup( info );
## Group([ (1,2)(3,4), (1,3,5) ])
## gap> AtlasGroup( info.identifier );
## Group([ (1,2)(3,4), (1,3,5) ])
## ]]></Example>
## <P/>
## In the groups returned by
## <Ref Func="AtlasGroup" Label="for various arguments"/>,
## the value of the attribute <Ref Attr="AtlasRepInfoRecord"/> is set.
## This information is used for example by
## <Ref Func="AtlasSubgroup" Label="for a group and a number"/>
## when this function is called with second argument a group created by
## <Ref Func="AtlasGroup" Label="for various arguments"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "AtlasGroup" );
#############################################################################
##
#F AtlasSubgroup( <gapname>[, <std>], <maxnr> )
#F AtlasSubgroup( <gapname>[, <std>], IsPermGroup[, true], <maxnr> )
#F AtlasSubgroup( <gapname>[, <std>], NrMovedPoints, <n>, <maxnr> )
#F AtlasSubgroup( <gapname>[, <std>], IsMatrixGroup[, true], <maxnr> )
#F AtlasSubgroup( <gapname>[, <std>][, Characteristic, <p>]
#F [, Dimension, <m>], <maxnr> )
#F AtlasSubgroup( <gapname>[, <std>][, Ring, <R>]
#F [, Dimension, <m>], <maxnr> )
#F AtlasSubgroup( <identifier>, <maxnr> )
#F AtlasSubgroup( <G>, <maxnr> )
##
## <#GAPDoc Label="AtlasSubgroup">
## <ManSection>
## <Heading>AtlasSubgroup</Heading>
## <Func Name="AtlasSubgroup" Arg='gapname[, std][, ...], maxnr'
## Label="for a group name (and various arguments) and a number"/>
## <Func Name="AtlasSubgroup" Arg='identifier, maxnr'
## Label="for an identifier record and a number"/>
## <Func Name="AtlasSubgroup" Arg='G, maxnr'
## Label="for a group and a number"/>
##
## <Returns>
## a group that satisfies the conditions, or <K>fail</K>.
## </Returns>
## <Description>
## The arguments of
## <Ref Func="AtlasSubgroup"
## Label="for a group name (and various arguments) and a number"/>,
## except the last argument <A>maxn</A>, are the same as for
## <Ref Func="AtlasGroup" Label="for various arguments"/>.
## If the &ATLAS; of Group Representations provides a straight line program
## for restricting representations of the group with name <A>gapname</A>
## (given w.r.t. the <A>std</A>-th standard generators)
## to the <A>maxnr</A>-th maximal subgroup
## and if a representation with the required properties is available,
## in the sense that calling
## <Ref Func="AtlasGroup" Label="for various arguments"/> with the same
## arguments except <A>maxnr</A> yields a group, then
## <Ref Func="AtlasSubgroup"
## Label="for a group name (and various arguments) and a number"/>
## returns the restriction of this representation to the <A>maxnr</A>-th
## maximal subgroup.
## <P/>
## In all other cases, <K>fail</K> is returned.
## <P/>
## Note that the conditions refer to the group and not to the subgroup.
## It may happen that in the restriction of a permutation representation
## to a subgroup, fewer points are moved,
## or that the restriction of a matrix representation turns out to be
## defined over a smaller ring.
## Here is an example.
## <P/>
## <Example><![CDATA[
## gap> g:= AtlasSubgroup( "A5", NrMovedPoints, 5, 1 );
## Group([ (1,5)(2,3), (1,3,5) ])
## gap> NrMovedPoints( g );
## 4
## ]]></Example>
## <P/>
## Alternatively, it is possible to enter exactly two arguments,
## the first being a record <A>identifier</A> as returned by
## <Ref Func="OneAtlasGeneratingSetInfo"/> or
## <Ref Func="AllAtlasGeneratingSetInfos"/>,
## or the <C>identifier</C> component of such a record,
## or a group <A>G</A> constructed with
## <Ref Func="AtlasGroup" Label="for an identifier record"/>.
## <P/>
## <Example><![CDATA[
## gap> info:= OneAtlasGeneratingSetInfo( "A5" );
## rec( groupname := "A5", id := "",
## identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ],
## isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,
## repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4",
## standardization := 1, transitivity := 3, type := "perm" )
## gap> AtlasSubgroup( info, 1 );
## Group([ (1,5)(2,3), (1,3,5) ])
## gap> AtlasSubgroup( info.identifier, 1 );
## Group([ (1,5)(2,3), (1,3,5) ])
## gap> AtlasSubgroup( AtlasGroup( "A5" ), 1 );
## Group([ (1,5)(2,3), (1,3,5) ])
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "AtlasSubgroup" );
#############################################################################
##
#F AtlasOfGroupRepresentationsUserParameters()
##
## <#GAPDoc Label="AtlasOfGroupRepresentationsShowUserParameters">
## <ManSection>
## <Func Name="AtlasOfGroupRepresentationsUserParameters" Arg=''/>
##
## <Description>
## This function returns a string that describes an overview of the current
## values of the user parameters introduced in this section.
## One can use <Ref Func="Print" BookName="ref"/> or
## <Ref Func="Pager" BookName="ref"/> for showing the overview.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "AtlasOfGroupRepresentationsUserParameters" );
#############################################################################
##
#E