#############################################################################
##
#W ctblothe.gd GAP 4 package CTblLib Thomas Breuer
##
## This file contains the declarations of functions for interfaces to
## other data formats of character tables.
##
## 1. Interface to CAS
## 2. Interface to MOC
## 3. Interface to GAP 3
## 4. Interface to the Cambridge format
## 5. Interface to the MAGMA display format
##
## <#GAPDoc Label="interfaces">
## This chapter describes data formats for character tables that can be read
## or created by &GAP;.
## Currently these are the formats used by
## <List>
## <Item>
## the &CAS; system (see <Ref Sect="sec:interface-cas"/>),
## </Item>
## <Item>
## the &MOC; system (see <Ref Sect="sec:interface-moc"/>),
## </Item>
## <Item>
## &GAP; 3 (see <Ref Sect="sec:interface-gap3"/>),
## </Item>
## <Item>
## the so-called <E>Cambridge format</E>
## (see <Ref Sect="sec:interface-cambridge"/>), and
## </Item>
## <Item>
## the <Package>MAGMA</Package> system
## (see <Ref Sect="sec:interface-magma"/>).
## </Item>
## </List>
## <#/GAPDoc>
##
#############################################################################
##
#T TODO:
##
#a MocData( <chi> )
#a MocInfo( <tbl> )
#o VirtualCharacterByMocData( <tbl>, <vector> )
#o CharacterByMocData( <tbl>, <vector> )
##
#############################################################################
##
## 1. Interface to CAS
##
## <#GAPDoc Label="interface_CAS">
## The interface to &CAS; (see <Cite Key="NPP84"/>) is thought
## just for printing the &CAS; data to a file.
## The function <Ref Func="CASString"/> is available mainly
## in order to document the data format.
## <E>Reading</E> &CAS; tables is not supported;
## note that the tables contained in the
## &CAS; Character Table Library have been migrated to
## &GAP; using a few <C>sed</C> scripts and <C>C</C> programs.
## <#/GAPDoc>
##
#############################################################################
##
#F CASString( <tbl> )
##
## <#GAPDoc Label="CASString">
## <ManSection>
## <Func Name="CASString" Arg="tbl"/>
##
## <Description>
## is a string that encodes the &CAS; library format
## of the character table <A>tbl</A>.
## This string can be printed to a file which then can be read into the
## &CAS; system using its <C>get</C> command (see <Cite Key="NPP84"/>).
## <P/>
## The used line length is the first entry in the list returned by
## <Ref Func="SizeScreen" BookName="ref"/>.
## <P/>
## Only the known values of the following attributes are used.
## <Ref Attr="ClassParameters" BookName="ref"/> (for partitions only),
## <Ref Attr="ComputedClassFusions" BookName="ref"/>,
## <Ref Attr="ComputedIndicators" BookName="ref"/>,
## <Ref Attr="ComputedPowerMaps" BookName="ref"/>,
## <Ref Attr="ComputedPrimeBlocks" BookName="ref"/>,
## <Ref Attr="Identifier" Label="for character tables" BookName="ref"/>,
## <Ref Attr="InfoText" BookName="ref"/>,
## <Ref Attr="Irr" BookName="ref"/>,
## <Ref Attr="OrdersClassRepresentatives" BookName="ref"/>,
## <Ref Attr="Size" BookName="ref"/>,
## <Ref Attr="SizesCentralizers" BookName="ref"/>.
## <P/>
## <Example><![CDATA[
## gap> Print( CASString( CharacterTable( "Cyclic", 2 ) ), "\n" );
## 'C2'
## 00/00/00. 00.00.00.
## (2,2,0,2,-1,0)
## text:
## (#computed using generic character table for cyclic groups#),
## order=2,
## centralizers:(
## 2,2
## ),
## reps:(
## 1,2
## ),
## powermap:2(
## 1,1
## ),
## characters:
## (1,1
## ,0:0)
## (1,-1
## ,0:0);
## /// converted from GAP
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "CASString" );
#############################################################################
##
## 2. Interface to MOC
##
## <#GAPDoc Label="interface_MOC">
## The interface to &MOC; (see <Cite Key="HJLP92"/>)
## can be used to print &MOC; input.
## Additionally it provides an alternative representation of (virtual)
## characters.
## <P/>
## The &MOC; 3 code of a <M>5</M> digit number
## in &MOC; 2 code is given by the following list.
## (Note that the code must contain only lower case letters.)
## <P/>
## <Verb>
## ABCD for 0ABCD
## a for 10000
## b for 10001 k for 20001
## c for 10002 l for 20002
## d for 10003 m for 20003
## e for 10004 n for 20004
## f for 10005 o for 20005
## g for 10006 p for 20006
## h for 10007 q for 20007
## i for 10008 r for 20008
## j for 10009 s for 20009
## tAB for 100AB
## uAB for 200AB
## vABCD for 1ABCD
## wABCD for 2ABCD
## yABC for 30ABC
## z for 31000
## </Verb>
## <P/>
## <E>Note</E> that any long number in &MOC; 2 format
## is divided into packages of length <M>4</M>,
## the first (!) one filled with leading zeros if necessary.
## Such a number with decimals <M>d_1, d_2, \ldots, d_{{4n+k}}</M>
## is the sequence
## <M>0 d_1 d_2 d_3 d_4 \ldots 0 d_{{4n-3}} d_{{4n-2}} d_{{4n-1}} d_{4n}
## d_{{4n+1}} \ldots d_{{4n+k}}</M>
## where <M>0 \leq k \leq 3</M>,
## the first digit of <M>x</M> is <M>1</M> if the number is positive
## and <M>2</M> if the number is negative,
## and then follow <M>(4-k)</M> zeros.
## <P/>
## Details about the &MOC; system are explained
## in <Cite Key="HJLP92"/>,
## a brief description can be found in <Cite Key="LP91"/>.
## <#/GAPDoc>
##
#############################################################################
##
#F MAKElb11( <listofns> )
##
## <#GAPDoc Label="MAKElb11">
## <ManSection>
## <Func Name="MAKElb11" Arg="listofns"/>
##
## <Description>
## For a list <A>listofns</A> of positive integers,
## <Ref Func="MAKElb11"/> prints field information for all number fields
## with conductor in this list.
## <P/>
## The output of <Ref Func="MAKElb11"/> is used by the &MOC; system;
## Calling <C>MAKElb11( [ 3 .. 189 ] )</C> will print something very similar
## to Richard Parker's file <F>lb11</F>.
## <P/>
## <Example><![CDATA[
## gap> MAKElb11( [ 3, 4 ] );
## 3 2 0 1 0
## 4 2 0 1 0
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "MAKElb11" );
#############################################################################
##
#F MOCTable( <gaptbl> )
#F MOCTable( <gaptbl>, <basicset> )
##
## <#GAPDoc Label="MOCTable">
## <ManSection>
## <Func Name="MOCTable" Arg="gaptbl[, basicset]"/>
##
## <Description>
## <Ref Func="MOCTable"/> returns the &MOC; table record of the
## &GAP; character table <A>gaptbl</A>.
## <P/>
## The one argument version can be used only if <A>gaptbl</A> is an
## ordinary (<M>G.0</M>) table.
## For Brauer (<M>G.p</M>) tables, one has to specify a basic set
## <A>basicset</A> of ordinary irreducibles.
## <A>basicset</A> must then be a list of positions of the basic set
## characters in the <Ref Attr="Irr" BookName="ref"/> list
## of the ordinary table of <A>gaptbl</A>.
## <P/>
## The result is a record that contains the information of <A>gaptbl</A>
## in a format similar to the &MOC; 3 format.
## This record can, e. g., easily be printed out or be used to print
## out characters using <Ref Func="MOCString"/>.
## <P/>
## The components of the result are
## <List>
## <Mark><C>identifier</C></Mark>
## <Item>
## the string <C>MOCTable( </C><M>name</M><C> )</C> where <M>name</M> is
## the <Ref Attr="Identifier" Label="for character tables" BookName="ref"/>
## value of <A>gaptbl</A>,
## </Item>
## <Mark><C>GAPtbl</C></Mark>
## <Item>
## <A>gaptbl</A>,
## </Item>
## <Mark><C>prime</C></Mark>
## <Item>
## the characteristic of the field (label <C>30105</C> in &MOC;),
## </Item>
## <Mark><C>centralizers</C></Mark>
## <Item>
## centralizer orders for cyclic subgroups (label <C>30130</C>)
## </Item>
## <Mark><C>orders</C></Mark>
## <Item>
## element orders for cyclic subgroups (label <C>30140</C>)
## </Item>
## <Mark><C>fieldbases</C></Mark>
## <Item>
## at position <M>i</M> the Parker basis of the number field generated
## by the character values of the <M>i</M>-th cyclic subgroup.
## The length of <C>fieldbases</C> is equal to the value of label
## <C>30110</C> in &MOC;.
## </Item>
## <Mark><C>cycsubgps</C></Mark>
## <Item>
## <C>cycsubgps[i] = j</C> means that class <C>i</C> of the &GAP; table
## belongs to the <C>j</C>-th cyclic subgroup of the &GAP; table,
## </Item>
## <Mark><C>repcycsub</C></Mark>
## <Item>
## <C>repcycsub[j] = i</C> means that class <C>i</C> of the &GAP; table
## is the representative of the <C>j</C>-th cyclic subgroup of the
## &GAP; table.
## <E>Note</E> that the representatives of &GAP; table and
## &MOC; table need not agree!
## </Item>
## <Mark><C>galconjinfo</C></Mark>
## <Item>
## a list <M>[ r_1, c_1, r_2, c_2, \ldots, r_n, c_n ]</M>
## which means that the <M>i</M>-th class of the &GAP; table is
## the <M>c_i</M>-th conjugate of the representative of
## the <M>r_i</M>-th cyclic subgroup on the &MOC; table.
## (This is used to translate back to &GAP; format,
## stored under label <C>30160</C>)
## </Item>
## <Mark><C>30170</C></Mark>
## <Item>
## (power maps) for each cyclic subgroup (except the trivial one)
## and each prime divisor of the representative order store four values,
## namely the number of the subgroup, the power,
## the number of the cyclic subgroup containing the image,
## and the power to which the representative must be raised to yield
## the image class.
## (This is used only to construct the <C>30230</C> power map/embedding
## information.)
## In <C>30170</C> only a list of lists (one for each cyclic subgroup)
## of all these values is stored, it will not be used by &GAP;.
## </Item>
## <Mark><C>tensinfo</C></Mark>
## <Item>
## tensor product information, used to compute the coefficients
## of the Parker base for tensor products of characters
## (label <C>30210</C> in &MOC;).
## For a field with vector space basis <M>(v_1, v_2, \ldots, v_n)</M>,
## the tensor product information of a cyclic subgroup in
## &MOC; (as computed by <C>fct</C>) is either <M>1</M>
## (for rational classes)
## or a sequence
## <Display Mode="M">
## n x_{1,1} y_{1,1} z_{1,1} x_{1,2} y_{1,2} z_{1,2}
## \ldots x_{1,m_1} y_{1,m_1} z_{1,m_1} 0 x_{2,1} y_{2,1}
## z_{2,1} x_{2,2} y_{2,2} z_{2,2} \ldots x_{2,m_2}
## y_{2,m_2} z_{2,m_2} 0 \ldots z_{n,m_n} 0
## </Display>
## which means that the coefficient of <M>v_k</M> in the product
## <Display Mode="M">
## \left( \sum_{i=1}^{n} a_i v_i \right)
## \left( \sum_{j=1}^{n} b_j v_j \right)
## </Display>
## is equal to
## <Display Mode="M">
## \sum_{i=1}^{m_k} x_{k,i} a_{y_{k,i}} b_{z_{k,i}} .
## </Display>
## On a &MOC; table in &GAP;,
## the <C>tensinfo</C> component is a list of lists,
## each containing exactly the sequence mentioned above.
## </Item>
## <Mark><C>invmap</C></Mark>
## <Item>
## inverse map to compute complex conjugate characters,
## label <C>30220</C> in &MOC;.
## </Item>
## <Mark><C>powerinfo</C></Mark>
## <Item>
## field embeddings for <M>p</M>-th symmetrizations,
## <M>p</M> a prime integer not larger than the largest element order,
## label <C>30230</C> in &MOC;.
## </Item>
## <Mark><C>30900</C></Mark>
## <Item>
## basic set of restricted ordinary irreducibles in the
## case of nonzero characteristic,
## all ordinary irreducibles otherwise.
## </Item>
## </List>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "MOCTable" );
#############################################################################
##
#F MOCString( <moctbl> )
#F MOCString( <moctbl>, <chars> )
##
## <#GAPDoc Label="MOCString">
## <ManSection>
## <Func Name="MOCString" Arg="moctbl[, chars]"/>
##
## <Description>
## Let <A>moctbl</A> be a &MOC; table record,
## as returned by <Ref Func="MOCTable"/>.
## <Ref Func="MOCString"/> returns a string describing the
## &MOC; 3 format of <A>moctbl</A>.
## <P/>
## If a second argument <A>chars</A> is specified,
## it must be a list of &MOC;
## format characters as returned by <Ref Func="MOCChars"/>.
## In this case, these characters are stored under label <C>30900</C>.
## If the second argument is missing then the basic set of ordinary
## irreducibles is stored under this label.
## <Example><![CDATA[
## gap> moca5:= MOCTable( CharacterTable( "A5" ) );
## rec( 30170 := [ [ ], [ 2, 2, 1, 1 ], [ 3, 3, 1, 1 ], [ 4, 5, 1, 1 ] ]
## ,
## 30900 := [ [ 1, 1, 1, 1, 0 ], [ 3, -1, 0, 0, -1 ],
## [ 3, -1, 0, 1, 1 ], [ 4, 0, 1, -1, 0 ], [ 5, 1, -1, 0, 0 ] ],
## GAPtbl := CharacterTable( "A5" ), centralizers := [ 60, 4, 3, 5 ],
## cycsubgps := [ 1, 2, 3, 4, 4 ],
## fieldbases :=
## [ CanonicalBasis( Rationals ), CanonicalBasis( Rationals ),
## CanonicalBasis( Rationals ),
## Basis( NF(5,[ 1, 4 ]), [ 1, E(5)+E(5)^4 ] ) ], fields := [ ],
## galconjinfo := [ 1, 1, 2, 1, 3, 1, 4, 1, 4, 2 ],
## identifier := "MOCTable(A5)",
## invmap := [ [ 1, 1, 0 ], [ 1, 2, 0 ], [ 1, 3, 0 ],
## [ 1, 4, 0, 1, 5, 0 ] ], orders := [ 1, 2, 3, 5 ],
## powerinfo :=
## [ ,
## [ [ 1, 1, 0 ], [ 1, 1, 0 ], [ 1, 3, 0 ],
## [ 1, 4, -1, 5, 0, -1, 5, 0 ] ],
## [ [ 1, 1, 0 ], [ 1, 2, 0 ], [ 1, 1, 0 ],
## [ 1, 4, -1, 5, 0, -1, 5, 0 ] ],,
## [ [ 1, 1, 0 ], [ 1, 2, 0 ], [ 1, 3, 0 ], [ 1, 1, 0, 0 ] ] ],
## prime := 0, repcycsub := [ 1, 2, 3, 4 ],
## tensinfo :=
## [ [ 1 ], [ 1 ], [ 1 ],
## [ 2, 1, 1, 1, 1, 2, 2, 0, 1, 1, 2, 1, 2, 1, -1, 2, 2, 0 ] ] )
## gap> str:= MOCString( moca5 );;
## gap> str{[1..68]};
## "y100y105ay110fey130t60edfy140bcdfy150bbbfcabbey160bbcbdbebecy170ccbb"
## gap> moca5mod3:= MOCTable( CharacterTable( "A5" ) mod 3, [ 1 .. 4 ] );;
## gap> MOCString( moca5mod3 ){ [ 1 .. 68 ] };
## "y100y105dy110edy130t60efy140bcfy150bbfcabbey160bbcbdbdcy170ccbbdfbby"
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "MOCString" );
#############################################################################
##
#F ScanMOC( <list> )
##
## <#GAPDoc Label="ScanMOC">
## <ManSection>
## <Func Name="ScanMOC" Arg="list"/>
##
## <Description>
## returns a record containing the information encoded in the list
## <A>list</A>.
## The components of the result are the labels that occur in <A>list</A>.
## If <A>list</A> is in &MOC; 2 format (10000-format),
## the names of components are 30000-numbers;
## if it is in &MOC; 3 format the names of components
## have <C>yABC</C>-format.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "ScanMOC" );
#############################################################################
##
#F GAPChars( <tbl>, <mocchars> )
##
## <#GAPDoc Label="GAPChars">
## <ManSection>
## <Func Name="GAPChars" Arg="tbl, mocchars"/>
##
## <Description>
## Let <A>tbl</A> be a character table or a &MOC;
## table record,
## and <A>mocchars</A> be either a list of &MOC; format
## characters
## (as returned by <Ref Func="MOCChars"/>)
## or a list of positive integers such as a record component encoding
## characters, in a record produced by <Ref Func="ScanMOC"/>.
## <P/>
## <Ref Func="GAPChars"/> returns translations of <A>mocchars</A> to &GAP;
## character values lists.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "GAPChars" );
#############################################################################
##
#F MOCChars( <tbl>, <gapchars> )
##
## <#GAPDoc Label="MOCChars">
## <ManSection>
## <Func Name="MOCChars" Arg="tbl, gapchars"/>
##
## <Description>
## Let <A>tbl</A> be a character table or a &MOC;
## table record,
## and <A>gapchars</A> be a list of (&GAP; format) characters.
## <Ref Func="MOCChars"/> returns translations of <A>gapchars</A>
## to &MOC; format.
## <Example><![CDATA[
## gap> scan:= ScanMOC( str );
## rec( y050 := [ 5, 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 1, 0, 0 ],
## y105 := [ 0 ], y110 := [ 5, 4 ], y130 := [ 60, 4, 3, 5 ],
## y140 := [ 1, 2, 3, 5 ], y150 := [ 1, 1, 1, 5, 2, 0, 1, 1, 4 ],
## y160 := [ 1, 1, 2, 1, 3, 1, 4, 1, 4, 2 ],
## y170 := [ 2, 2, 1, 1, 3, 3, 1, 1, 4, 5, 1, 1 ],
## y210 := [ 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 0, 1, 1, 2, 1, 2, 1, -1, 2,
## 2, 0 ], y220 := [ 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 4, 0, 1, 5, 0 ],
## y230 := [ 2, 1, 1, 0, 1, 1, 0, 1, 3, 0, 1, 4, -1, 5, 0, -1, 5, 0 ],
## y900 := [ 1, 1, 1, 1, 0, 3, -1, 0, 0, -1, 3, -1, 0, 1, 1, 4, 0, 1,
## -1, 0, 5, 1, -1, 0, 0 ] )
## gap> gapchars:= GAPChars( moca5, scan.y900 );
## [ [ 1, 1, 1, 1, 1 ], [ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ],
## [ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ], [ 4, 0, 1, -1, -1 ],
## [ 5, 1, -1, 0, 0 ] ]
## gap> mocchars:= MOCChars( moca5, gapchars );
## [ [ 1, 1, 1, 1, 0 ], [ 3, -1, 0, 0, -1 ], [ 3, -1, 0, 1, 1 ],
## [ 4, 0, 1, -1, 0 ], [ 5, 1, -1, 0, 0 ] ]
## gap> Concatenation( mocchars ) = scan.y900;
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "MOCChars" );
#############################################################################
##
## 3. Interface to GAP 3
##
## <#GAPDoc Label="interface_GAP3">
## The following functions are used to read and write character tables in
## &GAP; 3 format.
## <#/GAPDoc>
##
#############################################################################
##
#V GAP3CharacterTableData
##
## <#GAPDoc Label="GAP3CharacterTableData">
## <ManSection>
## <Var Name="GAP3CharacterTableData"/>
##
## <Description>
## This is a list of pairs,
## the first entry being the name of a component in a &GAP; 3
## character table and the second entry being the corresponding
## attribute name in &GAP; 4.
## The variable is used by <Ref Func="GAP3CharacterTableScan"/>
## and <Ref Func="GAP3CharacterTableString"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalVariable( "GAP3CharacterTableData",
"list of pairs [ <GAP 3 component>, <GAP 4 attribute> ]" );
#############################################################################
##
#F GAP3CharacterTableScan( <string> )
##
## <#GAPDoc Label="GAP3CharacterTableScan">
## <ManSection>
## <Func Name="GAP3CharacterTableScan" Arg="string"/>
##
## <Description>
## Let <A>string</A> be a string that contains the output of the
## &GAP; 3 function <C>PrintCharTable</C>.
## In other words, <A>string</A> describes a &GAP; record whose components
## define an ordinary character table object in &GAP; 3.
## <Ref Func="GAP3CharacterTableScan"/> returns the corresponding
## &GAP; 4 character table object.
## <P/>
## The supported record components are given by the list
## <Ref Var="GAP3CharacterTableData"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "GAP3CharacterTableScan" );
#############################################################################
##
#F GAP3CharacterTableString( <tbl> )
##
## <#GAPDoc Label="GAP3CharacterTableString">
## <ManSection>
## <Func Name="GAP3CharacterTableString" Arg="tbl"/>
##
## <Description>
## For an ordinary character table <A>tbl</A>,
## <Ref Func="GAP3CharacterTableString"/> returns
## a string that when read into &GAP; 3 evaluates to a character table
## corresponding to <A>tbl</A>.
## A similar format is printed by the &GAP; 3 function
## <C>PrintCharTable</C>.
## <P/>
## The supported record components are given by the list
## <Ref Var="GAP3CharacterTableData"/>.
## <P/>
## <Example><![CDATA[
## gap> tbl:= CharacterTable( "Alternating", 5 );;
## gap> str:= GAP3CharacterTableString( tbl );;
## gap> Print( str );
## rec(
## centralizers := [ 60, 4, 3, 5, 5 ],
## fusions := [ rec( map := [ 1, 3, 4, 7, 7 ], name := "Sym(5)" ) ],
## identifier := "Alt(5)",
## irreducibles := [
## [ 1, 1, 1, 1, 1 ],
## [ 4, 0, 1, -1, -1 ],
## [ 5, 1, -1, 0, 0 ],
## [ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ],
## [ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ]
## ],
## orders := [ 1, 2, 3, 5, 5 ],
## powermap := [ , [ 1, 1, 3, 5, 4 ], [ 1, 2, 1, 5, 4 ], , [ 1, 2, 3, 1, \
## 1 ] ],
## size := 60,
## text := "computed using generic character table for alternating groups\
## ",
## operations := CharTableOps )
## gap> scan:= GAP3CharacterTableScan( str );
## CharacterTable( "Alt(5)" )
## gap> TransformingPermutationsCharacterTables( tbl, scan );
## rec( columns := (), group := Group([ (4,5) ]), rows := () )
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "GAP3CharacterTableString" );
#############################################################################
##
## 4. Interface to the Cambridge format
##
## <#GAPDoc Label="interface_cambridge">
## The following functions deal with the so-called Cambridge format,
## in which the source data of the character tables in the
## &ATLAS; of Finite Groups <Cite Key="CCN85"/> and in the
## &ATLAS; of Brauer Characters <Cite Key="JLPW95"/> are stored.
## Each such table is stored on a file of its own.
## The line length is at most <M>78</M>,
## and each item of the table starts in a new line, behind one of the
## following prefixes.
## <P/>
## <List>
## <Mark><C>#23</C></Mark>
## <Item>
## a description and the name(s) of the simple group
## </Item>
## <Mark><C>#7</C></Mark>
## <Item>
## integers describing the column widths
## </Item>
## <Mark><C>#9</C></Mark>
## <Item>
## the symbols <C>;</C> and <C>@</C>, denoting columns between tables and
## columns that belong to conjugacy classes, respectively
## </Item>
## <Mark><C>#1</C></Mark>
## <Item>
## the symbol <C>|</C> in columns between tables, and centralizer orders
## otherwise
## </Item>
## <Mark><C>#2</C></Mark>
## <Item>
## the symbols <C>p</C> (in the first column only),
## <C>power</C> (in the second column only, which belongs to the class
## of the identity element), <C>|</C> in other columns between tables,
## and descriptions of the powers of classes otherwise
## </Item>
## <Mark><C>#3</C></Mark>
## <Item>
## the symbols <C>p'</C> (in the first column only),
## <C>part</C> (in the second column only, which belongs to the class
## of the identity element), <C>|</C> in other columns between tables,
## and descriptions of the <M>p</M>-prime parts of classes otherwise
## </Item>
## <Mark><C>#4</C></Mark>
## <Item>
## the symbols <C>ind</C> and <C>fus</C> in columns between tables,
## and class names otherwise
## </Item>
## <Mark><C>#5</C></Mark>
## <Item>
## either <C>|</C> or strings composed from the symbols <C>+</C>,
## <C>-</C>, <C>o</C>, and integers in columns where the lines
## starting with <C>#4</C> contain <C>ind</C>;
## the symbols <C>:</C>, <C>.</C>, <C>?</C> in columns where these
## lines contain <C>fus</C>;
## character values or <C>|</C> otherwise
## </Item>
## <Mark><C>#6</C></Mark>
## <Item>
## the symbols <C>|</C>, <C>ind</C>, <C>and</C>, and <C>fus</C>
## in columns between tables;
## the symbol <C>|</C> and element orders of preimage classes in downward
## extensions otherwise
## </Item>
## <Mark><C>#8</C></Mark>
## <Item>
## the last line of the data, may contain the date of the last change
## </Item>
## <Mark><C>#C</C></Mark>
## <Item>
## comments.
## </Item>
## </List>
## <#/GAPDoc>
##
#############################################################################
##
#F CambridgeMaps( <tbl> )
##
## <#GAPDoc Label="CambridgeMaps">
## <ManSection>
## <Func Name="CambridgeMaps" Arg="tbl"/>
##
## <Description>
## For a character table <A>tbl</A>, <Ref Func="CambridgeMaps"/> returns
## a record with the following components.
## <P/>
## <List>
## <Mark><C>names</C></Mark>
## <Item>
## a list of strings denoting class names,
## </Item>
## <Mark><C>power</C></Mark>
## <Item>
## a list of strings, the <M>i</M>-th entry encodes the <M>p</M>-th powers
## of the <M>i</M>-th class,
## for all prime divisors <M>p</M> of the group order,
## </Item>
## <Mark><C>prime</C></Mark>
## <Item>
## a list of strings, the <M>i</M>-th entry encodes the <M>p</M>-prime
## parts of the <M>i</M>-th class,
## for all prime divisors <M>p</M> of the group order.
## </Item>
## </List>
## <P/>
## The meaning of the entries of the lists is defined in
## <Cite Key="CCN85" Where="Chapter 7, Sections 3–5"/>).
## <P/>
## <Ref Func="CambridgeMaps"/> is used for example by
## <Ref Func="Display" Label="for a character table" BookName="ref"/>
## in the case that the <C>powermap</C> option has the value
## <C>"ATLAS"</C>.
## <P/>
## Note that the value of the <C>names</C> component may differ from the
## class names of the character table shown in the &ATLAS; of Finite Groups;
## an example is the character table of the group <M>J_1</M>.
## <P/>
## <Example><![CDATA[
## gap> CambridgeMaps( CharacterTable( "A5" ) );
## rec( names := [ "1A", "2A", "3A", "5A", "B*" ],
## power := [ "", "A", "A", "A", "A" ],
## prime := [ "", "A", "A", "A", "A" ] )
## gap> CambridgeMaps( CharacterTable( "A5" ) mod 2 );
## rec( names := [ "1A", "3A", "5A", "B*" ],
## power := [ "", "A", "A", "A" ], prime := [ "", "A", "A", "A" ] )
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "CambridgeMaps" );
#############################################################################
##
#F StringOfCambridgeFormat( <tblnames>[, <p>][: OmitDashedRows] )
#F StringOfCambridgeFormat( <simpname>[, <p>][: OmitDashedRows] )
#F StringOfCambridgeFormat( <tbls> ) . . . . . . . . backwards compatibility
##
## <#GAPDoc Label="StringOfCambridgeFormat">
## <ManSection>
## <Func Name="StringOfCambridgeFormat" Arg="tblnames[, p]"/>
##
## <Description>
## Let <A>tblnames</A> be a matrix of identifiers of ordinary character
## tables, which describe the bicyclic extensions of a simple group from the
## &ATLAS; of Finite Groups.
## The class fusions between the character tables must be stored on the
## tables.
## <P/>
## If the required information is available then
## <Ref Func="StringOfCambridgeFormat"/> returns a string that encodes an
## approximation of the Cambridge format file for the simple group in
## question (whose identifier occurs in the upper left corner of
## <A>tblnames</A>).
## Otherwise, that is, if some character table or class fusion is missing,
## <K>fail</K> is returned.
## <P/>
## If a prime integer <A>p</A> is given as a second argument then the
## result describes <A>p</A>-modular character tables, otherwise the
## ordinary character tables are described by the result.
## <P/>
## Differences to the original format may occur for irrational character
## values; the descriptions of these values have been chosen deliberately
## for the original files, it is not obvious how to compute these
## descriptions from the character tables in question.
## <P/>
## <Example><![CDATA[
## gap> str:= StringOfCambridgeFormat( [ [ "A5", "A5.2" ],
## > [ "2.A5", "2.A5.2" ] ] );;
## gap> Print( str );
## #23 ? A5
## #7 4 4 4 4 4 4 4 4 4 4 4
## #9 ; @ @ @ @ @ ; ; @ @ @
## #1 | 60 4 3 5 5 | | 6 2 3
## #2 p power A A A A | | A A AB
## #3 p' part A A A A | | A A AB
## #4 ind 1A 2A 3A 5A B* fus ind 2B 4A 6A
## #5 + 1 1 1 1 1 : ++ 1 1 1
## #5 + 3 -1 0 -b5 * . + 0 0 0
## #5 + 3 -1 0 * -b5 . | | | |
## #5 + 4 0 1 -1 -1 : ++ 2 0 -1
## #5 + 5 1 -1 0 0 : ++ 1 -1 1
## #6 ind 1 4 3 5 5 fus ind 2 8 6
## #6 | 2 | 6 10 10 | | | 8 6
## #5 - 2 0 -1 b5 * . - 0 0 0
## #5 - 2 0 -1 * b5 . | | | |
## #5 - 4 0 1 -1 -1 : oo 0 0 i3
## #5 - 6 0 0 1 1 : oo 0 i2 0
## #8
## gap> str:= StringOfCambridgeFormat( [ [ "A5", "A5.2" ],
## > [ "2.A5", "2.A5.2" ] ], 3 );;
## gap> Print( str );
## #23 A5 (Mod 3)
## #7 4 4 4 4 4 4 4 4 4
## #9 ; @ @ @ @ ; ; @ @
## #1 | 60 4 5 5 | | 6 2
## #2 p power A A A | | A A
## #3 p' part A A A | | A A
## #4 ind 1A 2A 5A B* fus ind 2B 4A
## #5 + 1 1 1 1 : ++ 1 1
## #5 + 3 -1 -b5 * . + 0 0
## #5 + 3 -1 * -b5 . | | |
## #5 + 4 0 -1 -1 : ++ 2 0
## #6 ind 1 4 5 5 fus ind 2 8
## #6 | 2 | 10 10 | | | 8
## #5 - 2 0 b5 * . - 0 0
## #5 - 2 0 * b5 . | | |
## #5 - 6 0 1 1 : oo 0 i2
## #8
## gap> StringOfCambridgeFormat( [ [ "L10(11)" ] ], 0 );
## fail
## ]]></Example>
## <P/>
## The global option <C>OmitDashedRows</C> can be used to control whether
## the two-line description of <Q>dashed</Q> row portions
## (concerning tables of, e. g., <M>2'.Sz(8)</M>)
## are omitted (value <K>true</K>) or shown (value <K>false</K>).
## The default is to show information about dashed row portions in the
## case of ordinary tables, and to omit this information for Brauer tables.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "StringOfCambridgeFormat" );
#############################################################################
##
## 5. Interface to the MAGMA display format
##
## <#GAPDoc Label="interface_magma">
## This interface is intended to convert character tables given in
## <Package>MAGMA</Package>'s (see <Cite Key="CP96"/>) display format
## into &GAP; character tables.
## <P/>
## The function <Ref Func="BosmaBase"/> is used for the translation of
## irrational values; this function may be of interest independent of the
## conversion of character tables.
## <#/GAPDoc>
##
#############################################################################
##
#F BosmaBase( <n> )
##
## <#GAPDoc Label="BosmaBase">
## <ManSection>
## <Func Name="BosmaBase" Arg="n"/>
##
## <Description>
## For a positive integer <A>n</A> that is not congruent to <M>2</M> modulo
## <M>4</M>, <Ref Func="BosmaBase"/> returns the list of exponents <M>i</M>
## for which <C>E(<A>n</A>)^</C><M>i</M> belongs to the canonical basis of
## the <A>n</A>-th cyclotomic field that is defined in
## <Cite Key="Bos90" Where="Section 5"/>.
## <P/>
## As a set, this basis is defined as follows.
## Let <M>P</M> denote the set of prime divisors of <A>n</A> and
## <A>n</A> <M>= \prod_{{p \in P}} n_p</M>.
## Let <M>e_l =</M> <C>E</C><M>(l)</M> for any positive integer <M>l</M>,
## and
## <M>\{ e_{{m_1}}^j \}_{{j \in J}} \otimes \{ e_{{m_2}}^k \}_{{k \in K}} =
## \{ e_{{m_1}}^j \cdot e_{{m_2}}^k \}_{{j \in J, k \in K}}</M>
## for any positive integers <M>m_1</M>, <M>m_2</M>.
## (This notation is the same as the one used in the description of
## <Ref Func="ZumbroichBase" BookName="ref"/>.)
## <P/>
## Then the basis is
## <Display Mode="M">
## B_n = \bigotimes_{{p \in P}} B_{{n_p}}
## </Display>
## where
## <Display Mode="M">
## B_{{n_p}} = \{ e_{{n_p}}^k; 0 \leq k \leq \varphi(n_p)-1 \};
## </Display>
## here <M>\varphi</M> denotes Euler's function,
## see <Ref Func="Phi" BookName="ref"/>.
## <P/>
## <M>B_n</M> consists of roots of unity, it is an integral basis
## (that is, exactly the integral elements in <M>&QQ;_n</M> have integral
## coefficients w.r.t. <M>B_n</M>,
## cf. <Ref Func="IsIntegralCyclotomic" BookName="ref"/>),
## and for any divisor <M>m</M> of <A>n</A> that is not congruent to
## <M>2</M> modulo <M>4</M>, <M>B_m</M> is a subset of <M>B_n</M>.
## <P/>
## Note that the list <M>l</M>, say, that is returned by
## <Ref Func="BosmaBase"/> is in general not a set.
## The ordering of the elements in <M>l</M> fits to the coefficient lists
## for irrational values used by <Package>MAGMA</Package>'s display format.
## <P/>
## <Example><![CDATA[
## gap> b:= BosmaBase( 8 );
## [ 0, 1, 2, 3 ]
## gap> b:= Basis( CF(8), List( b, i -> E(8)^i ) );
## Basis( CF(8), [ 1, E(8), E(4), E(8)^3 ] )
## gap> Coefficients( b, Sqrt(2) );
## [ 0, 1, 0, -1 ]
## gap> Coefficients( b, Sqrt(-2) );
## [ 0, 1, 0, 1 ]
## gap> b:= BosmaBase( 15 );
## [ 0, 5, 3, 8, 6, 11, 9, 14 ]
## gap> b:= List( b, i -> E(15)^i );
## [ 1, E(3), E(5), E(15)^8, E(5)^2, E(15)^11, E(5)^3, E(15)^14 ]
## gap> Coefficients( Basis( CF(15), b ), EB(15) );
## [ -1, -1, 0, 0, -1, -2, -1, -2 ]
## gap> BosmaBase( 48 );
## [ 0, 3, 6, 9, 12, 15, 18, 21, 16, 19, 22, 25, 28, 31, 34, 37 ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "BosmaBase" );
#############################################################################
##
#F GAPTableOfMagmaFile( <file>, <identifier> )
#F GAPTableOfMagmaFile( <str>, <identifier>, "string" )
##
## <#GAPDoc Label="GAPTableOfMagmaFile">
## <ManSection>
## <Func Name="GAPTableOfMagmaFile" Arg='file, identifier'
## Label="for a file"/>
## <Func Name="GAPTableOfMagmaFile" Arg='str, identifier[, "string"]'
## Label="for a string"/>
##
## <Description>
## In the first form,
## let <A>file</A> be the name of a file that contains a character table in
## <Package>MAGMA</Package>'s display format,
## and <A>identifier</A> be a string.
## <Ref Func="GAPTableOfMagmaFile" Label="for a file"/>
## returns the corresponding &GAP; character table,
## with <Ref Attr="Identifier" BookName="ref"/> value <A>identifier</A>.
## <P/>
## In the second form,
## <A>str</A> must be a string that describes the contents of a file
## as described for the first form, and the third argument must be the
## string <C>"string"</C>.
## <P/>
## <Example><![CDATA[
## gap> tmpdir:= DirectoryTemporary();;
## gap> file:= Filename( tmpdir, "magmatable" );;
## gap> str:= "\
## > Character Table of Group G\n\
## > --------------------------\n\
## > \n\
## > ---------------------------\n\
## > Class | 1 2 3 4 5\n\
## > Size | 1 15 20 12 12\n\
## > Order | 1 2 3 5 5\n\
## > ---------------------------\n\
## > p = 2 1 1 3 5 4\n\
## > p = 3 1 2 1 5 4\n\
## > p = 5 1 2 3 1 1\n\
## > ---------------------------\n\
## > X.1 + 1 1 1 1 1\n\
## > X.2 + 3 -1 0 Z1 Z1#2\n\
## > X.3 + 3 -1 0 Z1#2 Z1\n\
## > X.4 + 4 0 1 -1 -1\n\
## > X.5 + 5 1 -1 0 0\n\
## > \n\
## > Explanation of Character Value Symbols\n\
## > --------------------------------------\n\
## > \n\
## > # denotes algebraic conjugation, that is,\n\
## > #k indicates replacing the root of unity w by w^k\n\
## > \n\
## > Z1 = (CyclotomicField(5: Sparse := true)) ! [\n\
## > RationalField() | 1, 0, 1, 1 ]\n\
## > ";;
## gap> FileString( file, str );;
## gap> tbl:= GAPTableOfMagmaFile( file, "MagmaA5" );;
## gap> Display( tbl );
## MagmaA5
##
## 2 2 2 . . .
## 3 1 . 1 . .
## 5 1 . . 1 1
##
## 1a 2a 3a 5a 5b
## 2P 1a 1a 3a 5b 5a
## 3P 1a 2a 1a 5b 5a
## 5P 1a 2a 3a 1a 1a
##
## X.1 1 1 1 1 1
## X.2 3 -1 . A *A
## X.3 3 -1 . *A A
## X.4 4 . 1 -1 -1
## X.5 5 1 -1 . .
##
## A = -E(5)-E(5)^4
## = (1-Sqrt(5))/2 = -b5
## gap> tbl2:= GAPTableOfMagmaFile( str, "MagmaA5", "string" );;
## gap> Irr( tbl ) = Irr( tbl2 );
## true
## gap> str:= "\
## > Character Table of Group G\n\
## > --------------------------\n\
## > \n\
## > ------------------------------\n\
## > Class | 1 2 3 4 5 6\n\
## > Size | 1 1 1 1 1 1\n\
## > Order | 1 2 3 3 6 6\n\
## > ------------------------------\n\
## > p = 2 1 1 4 3 3 4\n\
## > p = 3 1 2 1 1 2 2\n\
## > ------------------------------\n\
## > X.1 + 1 1 1 1 1 1\n\
## > X.2 + 1 -1 1 1 -1 -1\n\
## > X.3 0 1 1 J-1-J-1-J J\n\
## > X.4 0 1 -1 J-1-J 1+J -J\n\
## > X.5 0 1 1-1-J J J-1-J\n\
## > X.6 0 1 -1-1-J J -J 1+J\n\
## > \n\
## > \n\
## > Explanation of Character Value Symbols\n\
## > --------------------------------------\n\
## > \n\
## > J = RootOfUnity(3)\n\
## > ";;
## gap> FileString( file, str );;
## gap> tbl:= GAPTableOfMagmaFile( file, "MagmaC6" );;
## gap> Display( tbl );
## MagmaC6
##
## 2 1 1 1 1 1 1
## 3 1 1 1 1 1 1
##
## 1a 2a 3a 3b 6a 6b
## 2P 1a 1a 3b 3a 3a 3b
## 3P 1a 2a 1a 1a 2a 2a
##
## X.1 1 1 1 1 1 1
## X.2 1 -1 1 1 -1 -1
## X.3 1 1 A /A /A A
## X.4 1 -1 A /A -/A -A
## X.5 1 1 /A A A /A
## X.6 1 -1 /A A -A -/A
##
## A = E(3)
## = (-1+Sqrt(-3))/2 = b3
## ]]></Example>
## <P/>
## The <Package>MAGMA</Package> output for the above two examples
## is obtained by the following commands.
## <P/>
## <Log>
## > G1 := Alt(5);
## > CT1 := CharacterTable(G1);
## > CT1;
## > G2:= CyclicGroup(6);
## > CT2:= CharacterTable(G2);
## > CT2;
## </Log>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "GAPTableOfMagmaFile" );
#############################################################################
##
#F UserPreference( "CTblLib", "MagmaPath" )
##
## <#GAPDoc Label="MagmaPath">
## <Subsection Label="subsect:magmapath">
## <Heading>User preference <C>MagmaPath</C></Heading>
## <Index Subkey="MagmaPath">user preferences</Index>
##
## This preference describes the path for calling &MAGMA;.
## An empty string (the default) means that &MAGMA; is not available.
## <P/>
## This preference is used by <Ref Func="CharacterTableComputedByMagma"/>.
## </Subsection>
## <#/GAPDoc>
##
## We cannot set a reasonable default path:
## Only a call to &MAGMA; would make sure that the program can be called,
## and we do not want to get an error message on startup if not.
##
DeclareUserPreference( rec( name:= "MagmaPath",
description:= [ "This preference describes the path for calling Magma. \
An empty string (the default) means that Magma is not available." ],
default:= "" ) );
#############################################################################
##
#F CharacterTableComputedByMagma( <G>, <identifier> )
##
## <#GAPDoc Label="CharacterTableComputedByMagma">
## <ManSection>
## <Func Name="CharacterTableComputedByMagma" Arg="G, identifier"/>
##
## <Description>
## For a permutation group <A>G</A> and a string <A>identifier</A>,
## <Ref Func="CharacterTableComputedByMagma"/> calls the &MAGMA; system
## for computing the character table of <A>G</A>,
## and converts the output into &GAP; format
## (see <Ref Func="GAPTableOfMagmaFile" Label="for a string"/>).
## The returned character table has the
## <Ref Attr="Identifier" BookName="ref"/> value <A>identifier</A>.
## <P/>
## If the &MAGMA; system is not available then <K>fail</K> is returned.
## The availability of &MAGMA; is determined by calling &MAGMA;
## where the path for this call is given by the user preference
## <C>MagmaPath</C> of the package &CTblLib;;
## if the value of this preference is empty
## or if &MAGMA; cannot be called via this path then
## &MAGMA; is regarded as not available.
## <P/>
## If the attribute <Ref Attr="ConjugacyClasses" BookName="ref"/>
## of <A>G</A> is set before the call of
## <Ref Func="CharacterTableComputedByMagma"/>
## then the columns of the returned character table fit to the
## conjugacy classes that are stored in <A>G</A>.
## <P/>
## <Example><![CDATA[
## gap> if CTblLib.IsMagmaAvailable() then
## > g:= MathieuGroup( 24 );
## > ccl:= ConjugacyClasses( g );
## > t:= CharacterTableComputedByMagma( g, "testM24" );
## > if t = fail then
## > Print( "#E Magma did not compute a character table.\n" );
## > elif ( not HasConjugacyClasses( t ) ) or
## > ( ConjugacyClasses( t ) <> ccl ) then
## > Print( "#E The conjugacy classes do not fit.\n" );
## > elif TransformingPermutationsCharacterTables( t,
## > CharacterTable( "M24" ) ) = fail then
## > Print( "#E Inconsistency of character tables?\n" );
## > fi;
## > fi;
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "CharacterTableComputedByMagma" );
#############################################################################
##
#E
