This chapter describes data formats for character tables that can be read or created by GAP. Currently these are the formats used by the CAS system (see Interface to the CAS System), the MOC system (see Interface to the MOC System), and GAP 3 (see Interface to GAP 3).
The interface to CAS is thought just for printing the CAS
data to a file.
The function CASString
is available mainly in order to document the
data format.
Reading CAS tables is not supported; note that the tables
contained in the CAS Character Table Library have been migrated to
GAP using a few sed
scripts and C
programs.
CASString(
tbl ) F
is a string that encodes the CAS library format of the character
table tbl.
This string can be printed to a file which then can be read into the
CAS system using its get
command (see NPP84).
The used line length is SizeScreen()[1]
(see SizeScreen in the GAP Reference Manual).
Only the known values of the following attributes are used.
ClassParameters
(for partitions only), ComputedClassFusions
,
ComputedPowerMaps
, Identifier
, InfoText
, Irr
,
ComputedPrimeBlocks
, ComputedIndicators
,
OrdersClassRepresentatives
, Size
, SizesCentralizers
.
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
The interface to MOC can be used to print MOC input. Additionally it provides an alternative representation of (virtual) characters.
The MOC 3 code of a 5 digit number in MOC 2 code is given by the following list. (Note that the code must contain only lower case letters.)
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
Note that any long number in MOC 2 format is divided into packages
of length 4, the first (!) one filled with leading zeros if necessary.
Such a number with decimals d1, d2, ¼, d4n+k is the sequence
|
HJLP92 explains details about the MOC system, a brief description can be found in LP91.
MAKElb11(
listofns ) F
MAKElb11
prints field information for all number fields with conductor
n where the positive integer n is in the list listofns.
The output of MAKElb11
is used by the MOC system.
MAKElb11( [ 3 .. 189 ] )
will print something very similar to
Richard Parker's file lb11
.
gap> MAKElb11( [ 3, 4 ] ); 3 2 0 1 0 4 2 0 1 0
MOCTable(
gaptbl ) F
MOCTable(
gaptbl,
basicset ) F
MOCTable
returns the MOC table record of the GAP character table
gaptbl.
The first form can be used only if gaptbl is an ordinary (G.0) table.
For Brauer (G.p) tables one has to specify a basic set basicset of
ordinary irreducibles.
basicset must be a list of positions of the basic set characters in the
Irr
list of the ordinary table of gaptbl.
The result is a record that contains the information of gaptbl
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 MOCString
(see MOCString).
The components of the result are
identifier
MOCTable(
name)
where name is the Identifier
value of gaptbl,
GAPtbl
prime
30105
in MOC),
centralizers
30130
)
orders
30140
)
fieldbases
fieldbases
is equal to the value of label 30110
in MOC.
cycsubgps
cycsubgps[i] = j
means that class i
of the GAP table
belongs to the j
-th cyclic subgroup of the GAP table,
repcycsub
repcycsub[j] = i
means that class i
of the GAP table
is the representative of the j
-th cyclic subgroup of the
GAP table.
Note that the representatives of GAP table and MOC table
need not agree!
galconjinfo
30160
)
30170
30230
power map/embedding
information.)
In 30170
only a list of lists (one for each cyclic subgroup)
of all these values is stored, it will not be used by GAP.
tensinfo
30210
in MOC).
For a field with vector space basis (v1,v2,¼,vn)
the tensor product information of a cyclic subgroup in MOC
(as computed by fct
) is either 1 (for rational classes)
or a sequence
|
|
|
tensinfo
component is
a list of lists, each containing exactly the sequence mentioned
above.
invmap
30220
in MOC.
powerinfo
30230
in MOC.
30900
MOCString(
moctbl ) F
MOCString(
moctbl,
chars ) F
Let moctbl be a MOC table record as returned by MOCTable
(see MOCTable).
MOCString
returns a string describing the MOC 3 format of moctbl.
If the second argument chars is specified, it must be a list of MOC
format characters as returned by MOCChars
(see MOCChars).
In this case, these characters are stored under label 30900
.
If the second argument is missing then the basic set of ordinary
irreducibles is stored under this label.
gap> moca5:= MOCTable( CharacterTable( "A5" ) ); rec( identifier := "MOCTable(A5)", prime := 0, fields := [ ], GAPtbl := CharacterTable( "A5" ), cycsubgps := [ 1, 2, 3, 4, 4 ], repcycsub := [ 1, 2, 3, 4 ], galconjinfo := [ 1, 1, 2, 1, 3, 1, 4, 1, 4, 2 ] , centralizers := [ 60, 4, 3, 5 ], orders := [ 1, 2, 3, 5 ], fieldbases := [ CanonicalBasis( Rationals ), CanonicalBasis( Rationals ), CanonicalBasis( Rationals ), Basis( NF(5,[ 1, 4 ]), [ 1, E(5)+E(5)^4 ] ) ], 30170 := [ [ ], [ 2, 2, 1, 1 ], [ 3, 3, 1, 1 ], [ 4, 5, 1, 1 ] ], tensinfo := [ [ 1 ], [ 1 ], [ 1 ], [ 2, 1, 1, 1, 1, 2, 2, 0, 1, 1, 2, 1, 2, 1, -1, 2, 2, 0 ] ], invmap := [ [ 1, 1, 0 ], [ 1, 2, 0 ], [ 1, 3, 0 ], [ 1, 4, 0, 1, 5, 0 ] ], 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 ] ] ], 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 ] ] ) gap> str:= MOCString( moca5 );; gap> str{[1..70]}; "y100y105ay110fey130t60edfy140bcdfy150bbbfcabbey160bbcbdbebecy170ccbbdd" gap> moca5mod3:= MOCTable( CharacterTable( "A5" ) mod 3, [ 1 .. 4 ] );; gap> MOCString( moca5mod3 ){ [ 1 .. 70 ] }; "y100y105dy110edy130t60efy140bcfy150bbfcabbey160bbcbdbdcy170ccbbdfbby21"
ScanMOC(
list ) F
returns a record containing the information encoded in the list list.
The components of the result are the labels that occur in list.
If list 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 yABC
-format.
GAPChars(
tbl,
mocchars ) F
Let tbl be a character table or a MOC table record,
and mocchars either a list of MOC format characters
(as returned by MOCChars
(see MOCChars)
or a list of positive integers such as a record component encoding
characters, in a record produced by ScanMOC
(see ScanMOC).
GAPChars
returns translations of mocchars to GAP character values
lists.
MOCChars(
tbl,
gapchars ) F
Let tbl be a character table or a MOC table record,
and gapchars a list of (GAP format) characters.
MOCChars
returns translations of gapchars to MOC format.
gap> scan:= ScanMOC( str ); rec( 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 ], y050 := [ 5, 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 1, 0, 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
The following functions are used to read and write character tables in GAP 3 format.
GAP3CharacterTableScan(
string ) F
Let string be a string that contains the output of the GAP 3
function PrintCharTable
.
In other words, string describes a GAP record whose components
define an ordinary character table object in GAP 3.
GAP3CharacterTableScan
returns the corresponding GAP 4
character table object.
The supported record components are given by the list
GAP3CharacterTableData
.
GAP3CharacterTableString(
tbl ) F
For an ordinary character table tbl, GAP3CharacterTableString
returns
a string that when read into GAP 3 evaluates to a character table
corresponding to tbl.
A similar format is printed by the GAP 3 function PrintCharTable
.
The supported record components are given by the list
GAP3CharacterTableData
.
gap> tbl:= CharacterTable( "Alternating", 5 );; gap> str:= GAP3CharacterTableString( tbl );; gap> Print( str ); rec( centralizers := [ 60, 4, 3, 5, 5 ], fusions := [ rec( name := "Sym(5)", map := [ 1, 3, 4, 7, 7 ] ) ], 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 := (), rows := (), group := Group([ (4,5) ]) )
GAP3CharacterTableData V
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 GAP3CharacterTableScan
(see GAP3CharacterTableScan) and GAP3CharacterTableString
(see GAP3CharacterTableString).
CTblLib manual