6 Interfaces to Other Data Formats for Character Tables

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 6.1),the

**MOC**system (see 6.2),**GAP**3 (see 6.3),the so-called

*Cambridge format*(see 6.4), andthe

**MAGMA**system (see 6.5).

The interface to **CAS** (see [NPP84]) is thought just for printing the **CAS** data to a file. The function `CASString`

(6.1-1) 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 ) | ( function ) |

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 the first entry in the list returned by `SizeScreen`

(Reference: SizeScreen).

Only the known values of the following attributes are used. `ClassParameters`

(Reference: ClassParameters) (for partitions only), `ComputedClassFusions`

(Reference: ComputedClassFusions), `ComputedIndicators`

(Reference: ComputedIndicators), `ComputedPowerMaps`

(Reference: ComputedPowerMaps), `ComputedPrimeBlocks`

(Reference: ComputedPrimeBlockss), `Identifier`

(Reference: Identifier for character tables), `InfoText`

(Reference: InfoText), `Irr`

(Reference: Irr), `OrdersClassRepresentatives`

(Reference: OrdersClassRepresentatives), `Size`

(Reference: Size), `SizesCentralizers`

(Reference: SizesCentralisers).

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** (see [HJLP]) 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 \(d_1, d_2, \ldots, d_{{4n+k}}\) is the sequence \(0 d_1 d_2 d_3 d_4 \ldots 0 d_{{4n-3}} d_{{4n-2}} d_{{4n-1}} d_{4n} d_{{4n+1}} \ldots d_{{4n+k}}\) where \(0 \leq k \leq 3\), the first digit of \(x\) is \(1\) if the number is positive and \(2\) if the number is negative, and then follow \((4-k)\) zeros.

Details about the **MOC** system are explained in [HJLP], a brief description can be found in [LP91].

`‣ MAKElb11` ( listofns ) | ( function ) |

For a list `listofns` of positive integers, `MAKElb11`

prints field information for all number fields with conductor in this list.

The output of `MAKElb11`

is used by the **MOC** system; Calling `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[, basicset] ) | ( function ) |

`MOCTable`

returns the **MOC** table record of the **GAP** character table `gaptbl`.

The one argument version 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 then be a list of positions of the basic set characters in the `Irr`

(Reference: 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`

(6.2-3).

The components of the result are

`identifier`

the string

`MOCTable(`

\(name\)`)`

where \(name\) is the`Identifier`

(Reference: Identifier for character tables) value of`gaptbl`,`GAPtbl`

`gaptbl`,`prime`

the characteristic of the field (label

`30105`

in**MOC**),`centralizers`

centralizer orders for cyclic subgroups (label

`30130`

)`orders`

element orders for cyclic subgroups (label

`30140`

)`fieldbases`

at position \(i\) the Parker basis of the number field generated by the character values of the \(i\)-th cyclic subgroup. The length of

`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`

a list \([ r_1, c_1, r_2, c_2, \ldots, r_n, c_n ]\) which means that the \(i\)-th class of the

**GAP**table is the \(c_i\)-th conjugate of the representative of the \(r_i\)-th cyclic subgroup on the**MOC**table. (This is used to translate back to**GAP**format, stored under label`30160`

)`30170`

(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

`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`

tensor product information, used to compute the coefficients of the Parker base for tensor products of characters (label

`30210`

in**MOC**). For a field with vector space basis \((v_1, v_2, \ldots, v_n)\), the tensor product information of a cyclic subgroup in**MOC**(as computed by`fct`

) is either \(1\) (for rational classes) or a sequence\[ n x_{1,1} y_{1,1} z_{1,1} x_{1,2} y_{1,2} z_{1,2} \ldots x_{1,m_1} y_{1,m_1} z_{1,m_1} 0 x_{2,1} y_{2,1} z_{2,1} x_{2,2} y_{2,2} z_{2,2} \ldots x_{2,m_2} y_{2,m_2} z_{2,m_2} 0 \ldots z_{n,m_n} 0 \]

which means that the coefficient of \(v_k\) in the product

\[ \left( \sum_{i=1}^{n} a_i v_i \right) \left( \sum_{j=1}^{n} b_j v_j \right) \]

is equal to

\[ \sum_{i=1}^{m_k} x_{k,i} a_{y_{k,i}} b_{z_{k,i}} . \]

On a

**MOC**table in**GAP**, the`tensinfo`

component is a list of lists, each containing exactly the sequence mentioned above.`invmap`

inverse map to compute complex conjugate characters, label

`30220`

in**MOC**.`powerinfo`

field embeddings for \(p\)-th symmetrizations, \(p\) a prime integer not larger than the largest element order, label

`30230`

in**MOC**.`30900`

basic set of restricted ordinary irreducibles in the case of nonzero characteristic, all ordinary irreducibles otherwise.

`‣ MOCString` ( moctbl[, chars] ) | ( function ) |

Let `moctbl` be a **MOC** table record, as returned by `MOCTable`

(6.2-2). `MOCString`

returns a string describing the **MOC** 3 format of `moctbl`.

If a second argument `chars` is specified, it must be a list of **MOC** format characters as returned by `MOCChars`

(6.2-6). 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( 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"

`‣ ScanMOC` ( list ) | ( function ) |

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 ) | ( function ) |

Let `tbl` be a character table or a **MOC** table record, and `mocchars` be either a list of **MOC** format characters (as returned by `MOCChars`

(6.2-6)) or a list of positive integers such as a record component encoding characters, in a record produced by `ScanMOC`

(6.2-4).

`GAPChars`

returns translations of `mocchars` to **GAP** character values lists.

`‣ MOCChars` ( tbl, gapchars ) | ( function ) |

Let `tbl` be a character table or a **MOC** table record, and `gapchars` be a list of (**GAP** format) characters. `MOCChars`

returns translations of `gapchars` to **MOC** format.

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

The following functions are used to read and write character tables in **GAP** 3 format.

`‣ GAP3CharacterTableScan` ( string ) | ( function ) |

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`

(6.3-3).

`‣ GAP3CharacterTableString` ( tbl ) | ( function ) |

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`

(6.3-3).

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 := () )

`‣ GAP3CharacterTableData` | ( global variable ) |

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`

(6.3-1) and `GAP3CharacterTableString`

(6.3-2).

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 [CCN+85] and in the **Atlas** of Brauer Characters [JLPW95] are stored. Each such table is stored on a file of its own. The line length is at most \(78\), and each item of the table starts in a new line, behind one of the following prefixes.

`#23`

a description and the name(s) of the simple group

`#7`

integers describing the column widths

`#9`

the symbols

`;`

and`@`

, denoting columns between tables and columns that belong to conjugacy classes, respectively`#1`

the symbol

`|`

in columns between tables, and centralizer orders otherwise`#2`

the symbols

`p`

(in the first column only),`power`

(in the second column only, which belongs to the class of the identity element),`|`

in other columns between tables, and descriptions of the powers of classes otherwise`#3`

the symbols

`p'`

(in the first column only),`part`

(in the second column only, which belongs to the class of the identity element),`|`

in other columns between tables, and descriptions of the \(p\)-prime parts of classes otherwise`#4`

the symbols

`ind`

and`fus`

in columns between tables, and class names otherwise`#5`

either

`|`

or strings composed from the symbols`+`

,`-`

,`o`

, and integers in columns where the lines starting with`#4`

contain`ind`

; the symbols`:`

,`.`

,`?`

in columns where these lines contain`fus`

; character values or`|`

otherwise`#6`

the symbols

`|`

,`ind`

,`and`

, and`fus`

in columns between tables; the symbol`|`

and element orders of preimage classes in downward extensions otherwise`#8`

the last line of the data, may contain the date of the last change

`#C`

comments.

`‣ CambridgeMaps` ( tbl ) | ( function ) |

For a character table `tbl`, `CambridgeMaps`

returns a record with the following components.

`names`

a list of strings denoting class names,

`power`

a list of strings, the \(i\)-th entry encodes the \(p\)-th powers of the \(i\)-th class, for all prime divisors \(p\) of the group order,

`prime`

a list of strings, the \(i\)-th entry encodes the \(p\)-prime parts of the \(i\)-th class, for all prime divisors \(p\) of the group order.

The meaning of the entries of the lists is defined in [CCN+85, Chapter 7, Sections 3–5]).

`CambridgeMaps`

is used for example by `Display`

(Reference: Display for a character table) in the case that the `powermap`

option has the value `"ATLAS"`

.

Note that the value of the `names`

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 \(J_1\).

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" ] )

`‣ StringOfCambridgeFormat` ( tblnames[, p] ) | ( function ) |

Let `tblnames` 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.

If the required information is available then `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 `tblnames`). Otherwise, that is, if some character table or class fusion is missing, `fail`

is returned.

If a prime integer `p` is given as a second argument then the result describes `p`-modular character tables, otherwise the ordinary character tables are described by the result.

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.

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

The global option `OmitDashedRows`

can be used to control whether the two-line description of "dashed" row portions (concerning tables of, e. g., \(2'.Sz(8)\)) are omitted (value `true`

) or shown (value `false`

). The default is to show information about dashed row portions in the case of ordinary tables, and to omit this information for Brauer tables.

This interface is intended to convert character tables given in **MAGMA**'s (see [CP96]) display format into **GAP** character tables.

The function `BosmaBase`

(6.5-1) is used for the translation of irrational values; this function may be of interest independent of the conversion of character tables.

`‣ BosmaBase` ( n ) | ( function ) |

For a positive integer `n` that is not congruent to \(2\) modulo \(4\), `BosmaBase`

returns the list of exponents \(i\) for which `E(`

\(i\) belongs to the canonical basis of the `n`)^`n`-th cyclotomic field that is defined in [Bos90, Section 5].

As a set, this basis is defined as follows. Let \(P\) denote the set of prime divisors of `n` and `n` \(= \prod_{{p \in P}} n_p\). Let \(e_l =\) `E`

\((l)\) for any positive integer \(l\), and \(\{ e_{{m_1}}^j \}_{{j \in J}} \otimes \{ e_{{m_2}}^k \}_{{k \in K}} = \{ e_{{m_1}}^j \cdot e_{{m_2}}^k \}_{{j \in J, k \in K}}\) for any positive integers \(m_1\), \(m_2\). (This notation is the same as the one used in the description of `ZumbroichBase`

(Reference: ZumbroichBase).)

Then the basis is

\[ B_n = \bigotimes_{{p \in P}} B_{{n_p}} \]

where

\[ B_{{n_p}} = \{ e_{{n_p}}^k; 0 \leq k \leq \varphi(n_p)-1 \}; \]

here \(\varphi\) denotes Euler's function, see `Phi`

(Reference: Phi).

\(B_n\) consists of roots of unity, it is an integral basis (that is, exactly the integral elements in \(ℚ_n\) have integral coefficients w.r.t. \(B_n\), cf. `IsIntegralCyclotomic`

(Reference: IsIntegralCyclotomic)), and for any divisor \(m\) of `n` that is not congruent to \(2\) modulo \(4\), \(B_m\) is a subset of \(B_n\).

Note that the list \(l\), say, that is returned by `BosmaBase`

is in general not a set. The ordering of the elements in \(l\) fits to the coefficient lists for irrational values used by **MAGMA**'s display format.

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 ]

`‣ GAPTableOfMagmaFile` ( file, identifier ) | ( function ) |

`‣ GAPTableOfMagmaFile` ( str, identifier[, "string"] ) | ( function ) |

In the first form, let `file` be the name of a file that contains a character table in **MAGMA**'s display format, and `identifier` be a string. `GAPTableOfMagmaFile`

returns the corresponding **GAP** character table, with `Identifier`

(Reference: Identifier for tables of marks) value `identifier`.

In the second form, `str` 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 `"string"`

.

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

The **MAGMA** output for the above two examples is obtained by the following commands.

> G1 := Alt(5); > CT1 := CharacterTable(G1); > CT1; > G2:= CyclicGroup(6); > CT2:= CharacterTable(G2); > CT2;

`‣ CharacterTableComputedByMagma` ( G, identifier ) | ( function ) |

For a permutation group `G` and a string `identifier`, `CharacterTableComputedByMagma`

calls the **MAGMA** system for computing the character table of `G`, and converts the output into **GAP** format (see `GAPTableOfMagmaFile`

(6.5-2)). The returned character table has the `Identifier`

(Reference: Identifier for tables of marks) value `identifier`.

If the **MAGMA** system is not available then `fail`

is returned. The availability of **MAGMA** is determined by calling **MAGMA** where the path for this call is given by the user preference `MagmaPath`

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.

If the attribute `ConjugacyClasses`

(Reference: ConjugacyClasses attribute) of `G` is set before the call of `CharacterTableComputedByMagma`

then the columns of the returned character table fit to the conjugacy classes that are stored in `G`.

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;

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