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SSL interfac.gd
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#############################################################################
##
#W interfac.gd GAP 4 package AtlasRep Thomas Breuer
##
## 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[, <bool>]]
#F [, NrMovedPoints, <n>]
#F [, IsTransitive[, <bool>]]
#F [, Transitivity, <n>]
#F [, IsPrimitive[, <bool>]]
#F [, RankAction, <n>]
#F [, IsMatrixGroup[, <bool>]]
#F [, Characteristic, <p>][, Dimension, <n>]
#F [, Ring, <R>]
#F [, Position, <n>]
#F [, Character, <chi>]
#F [, Identifier, <id>] )
#F DisplayAtlasInfo( <gapname>[, <std>][, "contents", <sources>]
#F [, IsPermGroup[, <bool>]]
#F [, NrMovedPoints, <n>]
#F [, IsTransitive[, <bool>]]
#F [, Transitivity, <n>]
#F [, IsPrimitive[, <bool>]]
#F [, RankAction, <n>]
#F [, IsMatrixGroup[, <bool>]]
#F [, Characteristic, <p>][, Dimension, <n>]
#F [, Ring, <R>]
#F [, Position, <n>]
#F [, Character, <chi>]
#F [, Identifier, <id>]
#F [, IsStraightLineProgram[, <bool>]] )
##
## <#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 &AtlasRep; package,
## for the given input.
## <P/>
## There are essentially three ways of calling this function.
## <List>
## <Item>
## If there is no argument or if the first argument is a list
## <A>listofnames</A> of strings that are &GAP; names of groups,
## <Ref Func="DisplayAtlasInfo"/> shows an overview of
## the known information.
## </Item>
## <Item>
## If the first argument is a string <A>gapname</A> that is a
## &GAP; name of a group,
## <Ref Func="DisplayAtlasInfo"/> shows an overview of the information
## that is available for this group.
## </Item>
## <Item>
## If the string <C>"contents"</C> is the only argument
## then the function shows which parts of the database are available;
## these are at least the <C>"core"</C> part, which means the data from
## the &ATLAS; of Group Representations, and the <C>"internal"</C> part,
## which means the data that are distributed with the &AtlasRep; package.
## Other parts can become available by calls to
## <Ref Func="AtlasOfGroupRepresentationsNotifyData"
## Label="for a local file describing private data"/>.
## Note that the shown numbers of locally available files depend on
## what has already been downloaded.
## </Item>
## </List>
## <P/>
## In each case,
## the information will be printed to the screen
## or will be fed into a pager,
## see Section <Ref Subsect="subsect:DisplayFunction"/>.
## An interactive alternative to <Ref Func="DisplayAtlasInfo"/> is the
## function <Ref Func="BrowseAtlasInfo" BookName="Browse"/>,
## see <Cite Key="Browse"/>.
## <P/>
## The following paragraphs describe the structure of the output in the
## two cases.
## Examples can be found in
## Section <Ref Subsect="sect:Examples for DisplayAtlasInfo"/>.
## <P/>
## Called without arguments, <Ref Func="DisplayAtlasInfo"/> shows a general
## overview for all groups.
## If some information is available for the group <M>G</M>, say,
## then one line is shown for <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"/>),
## if applicable followed by a string (by default a star <C>*</C>)
## indicating that at least one column refers to data not belonging to
## the core part of the database
## (see Section <Ref Subsect="subsect:AtlasRepMarkNonCoreData"/>).
## </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/>
## Called with a list <A>listofnames</A> of strings that are &GAP; names of
## some groups,
## <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 <M>sources</M> of the
## data may be given about which the overview is formed.
## See below for admissible values of <M>sources</M>.
## <P/>
## Called with a string <A>gapname</A> that is a &GAP; name of a group,
## <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, <M>id</M> is a (possibly empty) string.
## <P/>
## <List>
## <Mark><C>G <= Sym(</C><M>n</M><M>id</M><C>)</C></Mark>
## <Item>
## denotes a permutation representation of degree <M>n</M>,
## 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(</C><M>n</M><M>id</M>,<M>descr</M><C>)</C></Mark>
## <Item>
## denotes a matrix representation of dimension <M>n</M> over a
## coefficient ring described by <M>descr</M>,
## 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/</C><M>m</M><C>ℤ</C> for an integer <M>m</M>
## (denoting the ring of residues mod <M>m</M>);
## 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/>
## Below the representations,
## the programs available for <A>gapname</A> are listed.
## In each row of the overview, the entry in the first column is followed
## by a string (by default a star <C>*</C>) if the row refers to data
## not belonging to the core part of the database
## (see Section <Ref Subsect="subsect:AtlasRepMarkNonCoreData"/>).
## <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 <M>sources</M></Mark>
## <Item>
## for a string or a list of strings <M>sources</M>,
## restrict the data about which the overview is formed;
## if <M>sources</M> is the string <C>"core"</C> then only data from the
## &ATLAS; of Group Representations are considered,
## if <M>sources</M> is a string that denotes a data extension in the
## sense of a <C>dirid</C> argument of
## <Ref Func="AtlasOfGroupRepresentationsNotifyData"
## Label="for a local file describing private data"/> then
## only the data that belong to this data 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 <M>id</M></Mark>
## <Item>
## restrict to representations with <C>id</C> component in the
## list <M>id</M> (note that this component is itself a list, entering
## this list is not admissible),
## or satisfying the function <M>id</M>,
## </Item>
## <Mark><C>IsPermGroup</C> and <K>true</K> (or <K>false</K>)</Mark>
## <Item>
## restrict to permutation representations (or to representations that are
## not permutation representations),
## </Item>
## <Mark><C>NrMovedPoints</C> and <M>n</M></Mark>
## <Item>
## for a positive integer, a list of positive integers,
## or a property <M>n</M>,
## restrict to permutation representations of degree equal to <M>n</M>,
## or in the list <M>n</M>, or satisfying the function <M>n</M>,
## </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 a boolean value</Mark>
## <Item>
## restrict to transitive or intransitive permutation representations
## where this information is available (if the value <K>true</K> or
## <K>false</K> is given),
## or to representations for which this information is not available
## (if the value <K>fail</K> is given),
## </Item>
## <Mark><C>IsPrimitive</C> and a boolean value</Mark>
## <Item>
## restrict to primitive or imprimitive permutation representations
## where this information is available (if the value <K>true</K> or
## <K>false</K> is given),
## or to representations for which this information is not available
## (if the value <K>fail</K> is given),
## </Item>
## <Mark><C>Transitivity</C> and <M>n</M></Mark>
## <Item>
## for a nonnegative integer, a list of nonnegative integers,
## or a property <M>n</M>,
## restrict to permutation representations for which the information is
## available that the transitivity is equal to <M>n</M>,
## or is in the list <M>n</M>, or satisfies the function <M>n</M>;
## if <M>n</M> is <K>fail</K> then restrict to all permutation
## representations for which this information is not available,
## </Item>
## <Mark><C>RankAction</C> and <M>n</M></Mark>
## <Item>
## for a nonnegative integer, a list of nonnegative integers,
## or a property <M>n</M>,
## restrict to permutation representations for which the information is
## available that the rank is equal to <M>n</M>,
## or is in the list <M>n</M>, or satisfies the function <M>n</M>;
## if <M>n</M> is <K>fail</K> then restrict to all permutation
## representations for which this information is not available,
## </Item>
## <Mark><C>IsMatrixGroup</C> and <K>true</K> (or <K>false</K>)</Mark>
## <Item>
## restrict to matrix representations (or to representations that are
## not matrix representations),
## </Item>
## <Mark><C>Characteristic</C> and <M>p</M></Mark>
## <Item>
## for a prime integer, a list of prime integers, or a property <M>p</M>,
## restrict to matrix representations over fields of characteristic equal
## to <M>p</M>, or in the list <M>p</M>,
## or satisfying the function <M>p</M>
## (representations over residue class rings that are not fields can be
## addressed by entering <K>fail</K> as the value of <M>p</M>),
## </Item>
## <Mark><C>Dimension</C> and <M>n</M></Mark>
## <Item>
## for a positive integer, a list of positive integers,
## or a property <M>n</M>,
## restrict to matrix representations of dimension equal to <M>n</M>,
## or in the list <M>n</M>, or satisfying the function <M>n</M>,
## </Item>
## <Mark><C>Characteristic</C>, <M>p</M>, <C>Dimension</C>,
## and the string <C>"minimal"</C></Mark>
## <Item>
## for a prime integer <M>p</M>,
## restrict to faithful matrix representations over fields
## of characteristic <M>p</M> that have minimal dimension
## (if this information is available),
## </Item>
## <Mark><C>Ring</C> and <M>R</M></Mark>
## <Item>
## for a ring or a property <M>R</M>,
## restrict to matrix representations for which the information is
## available that the ring spanned by the matrix entries is contained
## in this ring or satisfies this property
## (note that the representation might be defined over a proper subring);
## if <M>R</M> is <K>fail</K> then restrict to all matrix representations
## for which this information is not available,
## </Item>
## <Mark><C>Ring</C>, <M>R</M>, <C>Dimension</C>,
## and the string <C>"minimal"</C></Mark>
## <Item>
## for a ring <M>R</M>, restrict to faithful matrix representations
## over this ring that have minimal dimension
## (if this information is available),
## </Item>
## <Mark><C>Character</C> and <M>chi</M></Mark>
## <Item>
## for a class function or a list of class functions <M>chi</M>,
## restrict to 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 representation),
## </Item>
## <Mark><C>Character</C> and <M>name</M></Mark>
## <Item>
## for a string <M>name</M>,
## restrict to representations for which the character is known to have
## this name,
## according to the information shown by <Ref Func="DisplayAtlasInfo"/>;
## if the characteristic is not specified then it defaults to zero,
## </Item>
## <Mark><C>Character</C> and <M>n</M></Mark>
## <Item>
## for a positive integer <M>n</M>,
## restrict to representations for which the character is known to be the
## <M>n</M>-th irreducible character in &GAP;'s library character table
## of the group in question;
## if the characteristic is not specified then it defaults to zero,
## </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"/>),
## and
## </Item>
## <Mark><C>IsStraightLineProgram</C> and <K>false</K></Mark>
## <Item>
## restrict to representations.
## </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.)
## </Description>
## </ManSection>
##
## <Subsection Label="sect:Examples for DisplayAtlasInfo">
## <Heading>Examples for DisplayAtlasInfo</Heading>
##
## Here are some examples how <Ref Func="DisplayAtlasInfo"/> can be called,
## and how its output can be interpreted.
## <P/>
## <Log><![CDATA[
## gap> DisplayAtlasInfo( "contents" );
## - AtlasRepAccessRemoteFiles: false
##
## - AtlasRepDataDirectory: /home/you/gap/pkg/atlasrep/
##
## ID | address, version, files
## ---------+------------------------------------------------
## core | http://atlas.math.rwth-aachen.de/Atlas/,
## | version 2019-04-08,
## | 10586 files locally available.
## ---------+------------------------------------------------
## internal | atlasrep/datapkg,
## | version 2019-05-06,
## | 276 files locally available.
## ---------+------------------------------------------------
## mfer | http://www.math.rwth-aachen.de/~mfer/datagens/,
## | version 2015-10-06,
## | 34 files locally available.
## ---------+------------------------------------------------
## ctblocks | ctblocks/atlas/,
## | version 2019-04-08,
## | 121 files locally available.
## ]]></Log>
## <P/>
## Note: The above output does not fit to the rest of the manual examples,
## since data extensions except <C>internal</C> have been removed at the
## beginning of Chapter <Ref Chap="chap:tutorial" Style="Number"/>.
## <P/>
## The output tells us that two data extensions have been notified
## in addition to the core data from the &ATLAS; of Group Representations
## and the (local) internal data distributed with the &AtlasRep; package.
## The files of the extension <C>mfer</C> must be downloaded before they
## can be read (but note that the access to remote files is disabled),
## and the files of the extension <C>ctblocks</C> are locally available
## in the <F>ctblocks/atlas</F> subdirectory of the &GAP; package directory.
## This table (in particular the numbers of locally available files)
## depends on your installation of the package and how many files you have
## already downloaded.
## <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 database provides
## <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.
## The star <C>*</C> in the first column of the row for <M>A_5</M> means
## that some of the available data do not belong to the core part of the
## database (see Section <Ref Subsect="subsect:AtlasRepMarkNonCoreData"/>).
## <P/>
## <Example><![CDATA[
## gap> DisplayAtlasInfo( [ "M11", "A5" ], NrMovedPoints, 11 );
## group | # | maxes | cl | cyc | out | fnd | chk | prs
## ------+---+-------+----+-----+-----+-----+-----+----
## M11 | 1 | 5 | + | + | | + | + | +
## ]]></Example>
## <P/>
## The given conditions restrict the overview to permutation representations
## on <M>11</M> points.
## The rows for all those groups are omitted for which no such
## representation is available, and the numbers of those representations are
## shown that satisfy the given conditions.
## In the above example, we see that no representation on <M>11</M> points
## is available for <M>A_5</M>, and exactly one such representation is
## available for <M>M_{11}</M>.
## <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) character 2a
## 9: G <= GL(2b,4) character 2b
## 10: G <= GL(3,5) character 3a
## 12: G <= GL(3a,9) character 3a
## 13: G <= GL(3b,9) character 3b
## 17: G <= GL(3a,Field([Sqrt(5)])) character 3a
## 18: G <= GL(3b,Field([Sqrt(5)])) character 3b
## gap> DisplayAtlasInfo( "A5", Characteristic, 0 );
## Representations for G = A5: (all refer to std. generators 1)
## ---------------------------
## 14: G <= GL(4,Z) character 4a
## 15: G <= GL(5,Z) character 5a
## 16: G <= GL(6,Z) character 3ab
## 17: G <= GL(3a,Field([Sqrt(5)])) character 3a
## 18: G <= GL(3b,Field([Sqrt(5)])) character 3b
## ]]></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) character 4a
## 8: G <= GL(2a,4) character 2a
## 12: G <= GL(3a,9) character 3a
## 17: G <= GL(3a,Field([Sqrt(5)])) character 3a
## ]]></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) character 4a
## 7: G <= GL(6,3) character 3ab
## 10: G <= GL(3,5) character 3a
## 11: G <= GL(5,5) character 5a
## 12: G <= GL(3a,9) character 3a
## 13: G <= GL(3b,9) character 3b
## gap> DisplayAtlasInfo( "A5", Dimension, IsPrimeInt );
## Representations for G = A5: (all refer to std. generators 1)
## ---------------------------
## 8: G <= GL(2a,4) character 2a
## 9: G <= GL(2b,4) character 2b
## 10: G <= GL(3,5) character 3a
## 11: G <= GL(5,5) character 5a
## 12: G <= GL(3a,9) character 3a
## 13: G <= GL(3b,9) character 3b
## 15: G <= GL(5,Z) character 5a
## 17: G <= GL(3a,Field([Sqrt(5)])) character 3a
## 18: G <= GL(3b,Field([Sqrt(5)])) character 3b
## gap> DisplayAtlasInfo( "A5", Ring, IsFinite and IsPrimeField );
## Representations for G = A5: (all refer to std. generators 1)
## ---------------------------
## 4: G <= GL(4a,2) character 4a
## 5: G <= GL(4b,2) character 2ab
## 6: G <= GL(4,3) character 4a
## 7: G <= GL(6,3) character 3ab
## 10: G <= GL(3,5) character 3a
## 11: G <= GL(5,5) character 5a
## ]]></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)
## --------------------
## - class repres.*
## - presentation
## - maxes (all 3):
## 1: A4
## 2: D10
## 3: S3
## - std. gen. checker:
## (check)
## (pres)
## ]]></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"/>.
## A straight line program for computing conjugacy class representatives
## is available, and the star <C>*</C> says that this program does not
## belong to the core part of the database
## (see Section <Ref Subsect="subsect:AtlasRepMarkNonCoreData"/>).
## </Subsection>
## <#/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 at least <A>repnr</A> representations for the group with &GAP; name
## <A>gapname</A> are available 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>contents</C></Mark>
## <Item>
## the identifier of the part of the database to which the generators
## belong, for example <C>"core"</C> or <C>"internal"</C>,
## </Item>
## <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,
## see Section <Ref Sect="sect:identifier component"/>;
## the value can be used as <C>identifier</C> argument of
## <Ref Func="AtlasGenerators"/>.
## </Item>
## <Mark><C>repname</C></Mark>
## <Item>
## a string that is an initial part of the filenames of the generators.
## </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>
## <Mark><C>type</C></Mark>
## <Item>
## a string that describes the type of the representation
## (<C>"perm"</C> for a permutation representation,
## <C>"matff"</C> for a matrix representation over a finite field,
## <C>"matint"</C> for a matrix representation over the ring of integers,
## <C>"matalg"</C> for a matrix representation over an algebraic number
## field).
## </Item>
## </List>
## <P/>
## Additionally, the following <E>describing components</E> may be available
## if they are known, and depending on the data type of the representation.
## <P/>
## <List>
## <Mark><C>size</C></Mark>
## <Item>
## the group order,
## </Item>
## <Mark><C>id</C></Mark>
## <Item>
## the distinguishing string as described for
## <Ref Func="DisplayAtlasInfo"/>,
## </Item>
## <Mark><C>charactername</C></Mark>
## <Item>
## a string that describes the character of the representation,
## </Item>
## <Mark><C>constituents</C></Mark>
## <Item>
## a list of positive integers denoting the positions of the irreducible
## constituents of the character of the representation,
## </Item>
## <Mark><C>p</C> (for permutation representations)</Mark>
## <Item>
## for the number of moved points,
## </Item>
## <Mark><C>dim</C> (for matrix representations)</Mark>
## <Item>
## the dimension of the matrices,
## </Item>
## <Mark><C>ring</C> (for matrix representations)</Mark>
## <Item>
## the ring generated by the matrix entries,
## </Item>
## <Mark><C>transitivity</C> (for permutation representations)</Mark>
## <Item>
## a nonnegative integer, see <Ref Oper="Transitivity" BookName="ref"/>,
## </Item>
## <Mark><C>orbits</C> (for intransitive permutation representations)</Mark>
## <Item>
## the sorted list of orbit lengths on the set of moved points,
## </Item>
## <Mark><C>rankAction</C> (for transitive permutation representations)</Mark>
## <Item>
## the number of orbits of the point stabilizer on the set of moved points,
## see <Ref Oper="RankAction" BookName="ref"/>,
## </Item>
## <Mark><C>stabilizer</C> (for transitive permutation representations)</Mark>
## <Item>
## a string that describes the structure of the point stabilizers,
## </Item>
## <Mark><C>isPrimitive</C> (for transitive permutation representations)</Mark>
## <Item>
## <K>true</K> if the point stabilizers are maximal subgroups,
## and <K>false</K> otherwise,
## </Item>
## <Mark><C>maxnr</C> (for primitive permutation representations)</Mark>
## <Item>
## the number of the class of maximal subgroups that contains the point
## stabilizers, w. r. t. the
## <Ref Attr="Maxes" BookName="ctbllib"/> list.
## </Item>
## </List>
## <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
## (for example after loading a package that provides some),
## <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( charactername := "1a+4a", constituents := [ 1, 4 ],
## contents := "core", 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( charactername := "2a", constituents := [ 2 ], contents := "core",
## 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( charactername := "3a", constituents := [ 2 ], contents := "core",
## 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 ],
## polynomial := [ -1, 1, 1 ], 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( charactername := "1a+4a", constituents := [ 1, 4 ],
## contents := "core", generators := [ (1,2)(3,4), (2,3)(4,5) ],
## groupname := "D10", id := "",
## identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5, 2 ],
## isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,
## repname := "A5G1-p5B0", repnr := 1, size := 10, stabilizer := "A4",
## standardization := 1, transitivity := 3, type := "perm" )
## 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>], "maxes", <maxnr>[, <std2>] )
#F AtlasProgramInfo( <gapname>[, <std>], "maxstd", <maxnr>, <vers>, <substd> )
#F AtlasProgramInfo( <gapname>[, <std>], "kernel", <factname> )
#F AtlasProgramInfo( <gapname>[, <std>], "classes" )
#F AtlasProgramInfo( <gapname>[, <std>], "cyclic" )
#F AtlasProgramInfo( <gapname>[, <std>], "cyc2ccl"[, <vers>] )
#F AtlasProgramInfo( <gapname>[, <std>], "automorphism", <autname> )
#F AtlasProgramInfo( <gapname>[, <std>], "check" )
#F AtlasProgramInfo( <gapname>[, <std>], "presentation" )
#F AtlasProgramInfo( <gapname>[, <std>], "find" )
#F AtlasProgramInfo( <gapname>, <std>, "restandardize", <std2> )
#F AtlasProgramInfo( <gapname>[, <std>], "other", <descr> )
##
## (same arguments as 'AtlasProgram')
##
## <#GAPDoc Label="AtlasProgramInfo">
## <ManSection>
## <Func Name="AtlasProgramInfo"
## Arg='gapname[, std][, "contents", sources][, "version", vers ], ...'/>
##
## <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 downloading files.
## 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, version := "1" )
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "AtlasProgramInfo" );
#############################################################################
##
#F AtlasProgram( <gapname>[, <std>][, "maxes"], <maxnr> )
#F AtlasProgram( <gapname>[, <std>], "maxes", <maxnr>[, <std2>] )
#F AtlasProgram( <gapname>[, <std>], "maxstd", <maxnr>, <vers>, <substd> )
#F AtlasProgram( <gapname>[, <std>], "kernel", <factname> )
#F AtlasProgram( <gapname>[, <std>], "classes" )
#F AtlasProgram( <gapname>[, <std>], "cyclic" )
#F AtlasProgram( <gapname>[, <std>], "cyc2ccl"[, <vers>] )
#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> )
##
## (Also the argument pairs '[ "contents", <sources> ]' and
## '[ "version", <v> ]' are supported.)
##
## <#GAPDoc Label="AtlasProgram">
## <ManSection>
## <Func Name="AtlasProgram"
## Arg='gapname[, std][, "contents", sources][, "version", vers ], ...'/>
## <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 database 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 arguments indicated by <A>...</A> (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/>
## If the optional arguments <C>"contents"</C> and <A>sources</A> are
## given then the latter must be either a string or a list of strings,
## with the same meaning as described for <Ref Func="DisplayAtlasInfo"/>.
## <P/>
## If the optional arguments <C>"version"</C> and <A>vers</A> are
## given then the latter must be either a number or a list of numbers,
## and only those straight line programs/decisions are considered
## whose version number fits to <A>vers</A>.
## <P/>
## The result record has at least the following components.
## <P/>
## <List>
## <Mark><C>groupname</C></Mark>
## <Item>
## the string <A>gapname</A>,
## </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>
## <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>version</C></Mark>
## <Item>
## the substring of the filename of the program that denotes the
## version of the program.
## </Item>
## </List>
## <P/>
## If the program computes generators of the restriction to a maximal
## subgroup then also the following components are present.
## <P/>
## <List>
## <Mark><C>size</C></Mark>
## <Item>
## the order of the maximal subgroup,
## </Item>
## <Mark><C>subgroupname</C></Mark>
## <Item>
## a string denoting a name of the maximal subgroup.
## </Item>
## </List>
## <P/>
## In the first form,
## the arguments indicated by <A>...</A> must be as follows.
## <P/>
## <List>
## <Mark>(the string <C>"maxes"</C> and) a positive integer <M>maxnr</M>
## </Mark>
## <Item>
## the required program computes generators of the <M>maxnr</M>-th
## maximal subgroup of the group with &GAP; name <M>gapname</M>.
## <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>the string <C>"maxes"</C>
## and two positive integers <M>maxnr</M> and <M>std2</M></Mark>
## <Item>
## the required program computes standard generators of the
## <M>maxnr</M>-th maximal subgroup of the group with &GAP; name
## <M>gapname</M>, w. r. t. the standardization <M>std2</M>.
## <P/>
## A prescribed <C>"version"</C> parameter refers to the straight line
## program for computing the restriction, not to the program for
## standardizing the result of the restriction.
## <P/>
## The meaning of the component <C>size</C> in the result, if present,
## is the same as in the previous case.
## </Item>
## <Mark>the string <C>"maxstd"</C> and three positive integers
## <M>maxnr</M>, <M>vers</M>, <M>substd</M></Mark>
## <Item>
## the required program computes standard generators of the
## <M>maxnr</M>-th maximal subgroup of the group with &GAP; name
## <M>gapname</M> w. r. t. standardization <M>substd</M>;
## in this case, the inputs of the program are <E>not</E> standard
## generators of the group with &GAP; name <M>gapname</M>
## but the outputs of the straight line program with version <M>vers</M>
## for computing generators of its <M>maxnr</M>-th maximal subgroup.
## </Item>
## <Mark>the string <C>"kernel"</C> and a string <M>factname</M></Mark>
## <Item>
## the required program computes generators of the kernel of an
## epimorphism from <M>G</M> to a group with &GAP; name <M>factname</M>.
## <Index Subkey="for normal subgroups">straight line program</Index>
## <Index Subkey="for kernels of epimorphisms">straight line program</Index>
## </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 string <C>"cyc2ccl"</C> (and the string <M>vers</M>)</Mark>
## <Item>
## the required program computes representatives of conjugacy classes
## of elements from representatives of generators of maximally cyclic
## subgroups of <M>G</M>.
## Thus the inputs are the outputs of the program of type <C>"cyclic"</C>
## whose version is <M>vers</M>.
## </Item>
## <Mark>the strings <C>"cyc2ccl"</C>, <M>vers1</M>, <C>"version"</C>, <M>vers2</M></Mark>
## <Item>
## the required program computes representatives of conjugacy classes
## of elements from representatives of generators of maximally cyclic
## subgroups of <M>G</M>,
## where the inputs are the outputs of the program of type <C>"cyclic"</C>
## whose version is <M>vers1</M> and the required program itself has
## version <M>vers2</M>.
## </Item>
## <Mark>the strings <C>"automorphism"</C> and <M>autname</M></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 <M>autname</M> 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 <M>std2</M></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
## <M>std2</M>-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 <M>descr</M></Mark>
## <Item>
## <Index Subkey="free format">straight line program</Index>
## the required program is described by <M>descr</M>.
## </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", version := "1" )
## gap> StringOfResultOfStraightLineProgram( prog.program, [ "a", "b" ] );
## "[ a, bbab ]"
## gap> gens1:= AtlasGenerators( "A5", 1 );
## rec( charactername := "1a+4a", constituents := [ 1, 4 ],
## contents := "core", 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,
## version := "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 database contains at least one representation for <M>G</M> with
## the required properties
## then <Ref Func="OneAtlasGeneratingSetInfo"/> returns a record <M>r</M>
## 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,
## and an additional component <C>givenRing</C> is present if <C>Ring</C>
## is one of the arguments in the function call.
## <P/>
## The information in <C>givenRing</C> can be used later to construct
## the matrices over the prescribed ring.
## Note that this ring may be for example a domain constructed with
## <Ref Func="AlgebraicExtension" BookName="ref"/> instead of a field of
## cyclotomics or of a finite field constructed with
## <Ref Func="GF" BookName="ref"/>.
## <P/>
## The component <C>identifier</C> of <M>r</M> 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 preference
## <C>AtlasRepAccessRemoteFiles</C> has the value <K>true</K>
## (see Section <Ref Subsect="subsect:AtlasRepAccessRemoteFiles"/>),
## <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( charactername := "1a+4a", constituents := [ 1, 4 ],
## contents := "core", 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( charactername := "1a+4a", constituents := [ 1, 4 ],
## contents := "core", 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 database does not store this imprimitive representation (cf.
## Section <Ref Sect="sect:Accessing vs. Constructing Representations"/>).
## <P/>
## We continue this example.
## Next we access matrix representations of <M>A_5</M>.
## <P/>
## <Example><![CDATA[
## gap> info:= OneAtlasGeneratingSetInfo( "A5", IsMatrixGroup, true );
## rec( charactername := "4a", constituents := [ 4 ], contents := "core",
## 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( charactername := "4a", constituents := [ 4 ], contents := "core",
## 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> info2:= OneAtlasGeneratingSetInfo( "A5", Ring, GF(2) );;
## gap> info.identifier = info2.identifier;
## true
## gap> OneAtlasGeneratingSetInfo( "A5", Characteristic, [2,5], Dimension, 2 );
## rec( charactername := "2a", constituents := [ 2 ], contents := "core",
## 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( charactername := "4a", constituents := [ 4 ], contents := "core",
## 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( charactername := "4a", constituents := [ 4 ], contents := "core",
## 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> info2:= OneAtlasGeneratingSetInfo( "A5", Ring, CF(37) );;
## gap> info = info2;
## false
## gap> Difference( RecNames( info2 ), RecNames( info ) );
## [ "givenRing" ]
## gap> info2.givenRing;
## CF(37)
## gap> OneAtlasGeneratingSetInfo( "A5", Ring, Integers mod 77 );
## fail
## gap> info:= OneAtlasGeneratingSetInfo( "A5", Ring, CF(5), Dimension, 3 );
## rec( charactername := "3a", constituents := [ 2 ], contents := "core",
## dim := 3, givenRing := CF(5), groupname := "A5", id := "a",
## identifier := [ "A5", "A5G1-Ar3aB0.g", 1, 3 ],
## polynomial := [ -1, 1, 1 ], repname := "A5G1-Ar3aB0", repnr := 17,
## ring := NF(5,[ 1, 4 ]), size := 60, standardization := 1,
## type := "matalg" )
## gap> gens:= AtlasGenerators( info );
## rec( charactername := "3a", constituents := [ 2 ], contents := "core",
## 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 ] ] ],
## givenRing := CF(5), groupname := "A5", id := "a",
## identifier := [ "A5", "A5G1-Ar3aB0.g", 1, 3 ],
## polynomial := [ -1, 1, 1 ], repname := "A5G1-Ar3aB0", repnr := 17,
## ring := NF(5,[ 1, 4 ]), size := 60, standardization := 1,
## type := "matalg" )
## gap> gens2:= AtlasGenerators( info.identifier );;
## gap> Difference( RecNames( gens ), RecNames( gens2 ) );
## [ "givenRing" ]
## 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( charactername := "1a+4a", constituents := [ 1, 4 ],
## contents := "core", groupname := "A5", id := "",
--> --------------------
--> maximum size reached
--> --------------------
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2026-03-28
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