%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %A paramaps.g GAP documentation Thomas Breuer %% %A @(#)$Id: paramaps.tex,v 3.12 1994/06/10 02:50:14 vfelsch Rel $ %% %Y Copyright 1990-1992, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany %% %H $Log: paramaps.tex,v $ %H Revision 3.12 1994/06/10 02:50:14 vfelsch %H updated examples %H %H Revision 3.11 1994/06/03 08:57:20 mschoene %H changed a few things to avoid LaTeX warnings %H %H Revision 3.10 1994/05/31 11:35:04 vfelsch %H long line adjusted %H %H Revision 3.9 1994/05/31 10:52:49 vfelsch %H fixed overfull box %H %H Revision 3.8 1994/05/25 12:51:56 sam %H improved description of 'SubgroupFusions' %H %H Revision 3.7 1993/02/19 10:48:42 gap %H adjustments in line length and spelling %H %H Revision 3.6 1993/02/18 14:48:13 felsch %H long line adjusted %H %H Revision 3.5 1993/02/15 15:57:37 felsch %H examples fixed %H %H Revision 3.3 1992/02/13 15:05:56 sam %H renamed 'MatrixAutomorphisms' to 'MatAutomorphisms' %H %H Revision 3.2 1992/01/21 17:13:50 sam %H added description of 'CheckPermChar', %H changed 'InitFusion' and examples appropriately %H %H Revision 3.1 1991/12/27 16:46:58 sam %H initial revision under RCS %H %% \Chapter{Maps and Parametrized Maps}\index{maps}\index{parametrized maps} In this chapter, first the data structure of (parametrized) maps is introduced (see "More about Maps and Parametrized Maps"). Then a description of several functions which mainly deal with parametrized maps follows; these are basic operations with paramaps (see "CompositionMaps", "InverseMap", "Indirected", "ProjectionMap", "Parametrized", "ContainedMaps", "UpdateMap", "CommutativeDiagram", "TransferDiagram"), functions which inform about ambiguity with respect to a paramap (see "Indeterminateness", "PrintAmbiguity"), functions used for the construction of powermaps and subgroup fusions (see "Powermap", "SubgroupFusions" and their subroutines "InitPowermap", "Congruences", "ConsiderKernels", "ConsiderSmallerPowermaps", "PowermapsAllowedBySymmetrisations", "InitFusion", "CheckPermChar", "CheckFixedPoints", "TestConsistencyMaps", "ConsiderTableAutomorphisms", "FusionsAllowedByRestrictions", "OrbitPowermaps", "OrbitFusions", "RepresentativesPowermaps", "RepresentativesFusions", "Powmap") and the function "ElementOrdersPowermap". %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{More about Maps and Parametrized Maps} \index{map!parametrized}\index{paramap}\index{class functions} Besides the characters, *powermaps* are an important part of a character table. Often their computation is not easy, and in general they cannot be obtained from the matrix of irreducible characters, so it is useful to store them on the table. If not only a single table is considered but different tables of groups and subgroups are used, also *subgroup fusion maps* must be known to get informations about the embedding or simply to induce or restrict characters. These are examples of class functions which are called *maps* for short; in \GAP, maps are lists\:\ Characters are maps, the lists of element orders, centralizer orders, classlengths are maps, and for a permutation of classes, 'ListPerm( )' is a map. When maps are constructed, in most cases one only knows that the image of any class is contained in a set of possible images, e.g. that the image of a class under a subgroup fusion is in the set of all classes with the same element order. Using further informations, like centralizer orders, powermaps and the restriction of characters, the sets of possible images can be diminished. In many cases, at the end the images are uniquely determined. For this, many functions do not only work with maps but with *parametrized maps* (or short paramaps)\:\ These are lists whose entries are either the images themselves (i.e. integers for fusion maps, powermaps, element orders etc.\ and cyclotomics for characters) or lists of possible images. Thus maps are special paramaps. A paramap can be identified with the set of all maps with '[i] = [i]' or '[i]' contained in '[i]'; we say that is contained in then. The composition of two paramaps is defined as the paramap that contains all compositions of elements of the paramaps. For example, the indirection of a character by a parametrized subgroup fusion map is the parametrized character that contains all possible restrictions of that character. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{CompositionMaps}\index{map!composition} 'CompositionMaps( , )'\\ 'CompositionMaps( , , )' For parametrized maps and where '[i]' is a bound position or a set of bound positions in , 'CompositionMaps( , )' is a parametrized map with image 'CompositionMaps( , , )' at position . If '[ ]' is unique, we have \[ 'CompositionMaps( , , ) = [ [ ] ]', \] otherwise it is the union of '[i]' for 'i' in '[ ]'. | gap> map1:= [ 1, [ 2, 3, 4 ], [ 4, 5 ], 1 ];; gap> map2:= [ [ 1, 2 ], 2, 2, 3, 3 ];; gap> CompositionMaps( map2, map1 ); CompositionMaps( map1, map2 ); [ [ 1, 2 ], [ 2, 3 ], 3, [ 1, 2 ] ] [ [ 1, 2, 3, 4 ], [ 2, 3, 4 ], [ 2, 3, 4 ], [ 4, 5 ], [ 4, 5 ] ]| *Note*\:\ If you want to get indirections of characters which contain unknowns (see chapter "Unknowns") instead of sets of possible values, use "Indirected" 'Indirected'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{InverseMap}\index{map!inverse of a} 'InverseMap( )' 'InverseMap( )[i]' is the unique preimage or the set of all preimages of 'i' under , if there are any; otherwise it is unbound. (We have 'CompositionMaps( , InverseMap( ) )' the identity map.) | gap> t:= CharTable( "2.A5" );; f:= CharTable( "A5" );; gap> fus:= GetFusionMap( t, f ); # the factor fusion map [ 1, 1, 2, 3, 3, 4, 4, 5, 5 ] gap> inverse:= InverseMap( fus ); [ [ 1, 2 ], 3, [ 4, 5 ], [ 6, 7 ], [ 8, 9 ] ] gap> CompositionMaps( fus, inverse ); [ 1, 2, 3, 4, 5 ] gap> t.powermap[2]; [ 1, 1, 2, 4, 4, 8, 8, 6, 6 ] # transfer a powermap up to the factor group\: gap> CompositionMaps( fus, CompositionMaps( last, inverse ) ); [ 1, 1, 3, 5, 4 ] # is equal to 'f.powermap[2]' # transfer a powermap down to the group\: gap> CompositionMaps( inverse, CompositionMaps( last, fus ) ); [ [ 1, 2 ], [ 1, 2 ], [ 1, 2 ], [ 4, 5 ], [ 4, 5 ], [ 8, 9 ], [ 8, 9 ], [ 6, 7 ], [ 6, 7 ] ] # contains 't.powermap[2]'| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{ProjectionMap} 'ProjectionMap( )' For each image 'i' under the (necessarily *not* parametrized) map , 'ProjectionMap( )[]' is the smallest preimage of . (We have 'CompositionMaps( , ProjectionMap( ) )' the identity map.) | gap> ProjectionMap( [1,1,1,2,2,2,3,4,5,5,5,6,6,6,7,7,7] ); [ 1, 4, 7, 8, 9, 12, 15 ]| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Parametrized}\index{map!parametrize} 'Parametrized( )' returns the parametrized cover of , i.e.\ the parametrized map with smallest indeterminateness that contains all maps in . 'Parametrized' is the inverse function of "ContainedMaps" in the sense that 'Parametrized( ContainedMaps( ) ) = '. | gap> Parametrized( [ [ 1, 3, 4, 6, 8, 10, 11, 11, 15, 14 ], > [ 1, 3, 4, 6, 8, 10, 11, 11, 14, 15 ], > [ 1, 3, 4, 7, 8, 10, 12, 12, 15, 14 ], > [ 1, 3, 4, 7, 8, 10, 12, 12, 14, 15 ] ] ); [ 1, 3, 4, [ 6, 7 ], 8, 10, [ 11, 12 ], [ 11, 12 ], [ 14, 15 ], [ 14, 15 ] ]| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{ContainedMaps} 'ContainedMaps( )' returns the set of all maps contained in the parametrized map . 'ContainedMaps' is the inverse function of "Parametrized" in the sense that 'Parametrized( ContainedMaps( ) ) = '. | gap> ContainedMaps( [ 1, 3, 4, [ 6, 7 ], 8, 10, [ 11, 12 ], [ 11, 12 ], > 14, 15 ] ); [ [ 1, 3, 4, 6, 8, 10, 11, 11, 14, 15 ], [ 1, 3, 4, 6, 8, 10, 11, 12, 14, 15 ], [ 1, 3, 4, 6, 8, 10, 12, 11, 14, 15 ], [ 1, 3, 4, 6, 8, 10, 12, 12, 14, 15 ], [ 1, 3, 4, 7, 8, 10, 11, 11, 14, 15 ], [ 1, 3, 4, 7, 8, 10, 11, 12, 14, 15 ], [ 1, 3, 4, 7, 8, 10, 12, 11, 14, 15 ], [ 1, 3, 4, 7, 8, 10, 12, 12, 14, 15 ] ]| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{UpdateMap} 'UpdateMap( , , )' improves the paramap using that is the (possibly parametrized) indirection of the character by . | gap> s:= CharTable( "S4(4).2" );; he:= CharTable( "He" );; gap> fus:= InitFusion( s, he ); [ 1, 2, 2, [ 2, 3 ], 4, 4, [ 7, 8 ], [ 7, 8 ], 9, 9, 9, [ 10, 11 ], [ 10, 11 ], 18, 18, 25, 25, [ 26, 27 ], [ 26, 27 ], 2, [ 6, 7 ], [ 6, 7 ], [ 6, 7, 8 ], 10, 10, 17, 17, 18, [ 19, 20 ], [ 19, 20 ] ] gap> Filtered( s.irreducibles, x -> x[1] = 50 ); [ [ 50, 10, 10, 2, 5, 5, -2, 2, 0, 0, 0, 1, 1, 0, 0, 0, 0, -1, -1, 10, 2, 2, 2, 1, 1, 0, 0, 0, -1, -1 ], [ 50, 10, 10, 2, 5, 5, -2, 2, 0, 0, 0, 1, 1, 0, 0, 0, 0, -1, -1, -10, -2, -2, -2, -1, -1, 0, 0, 0, 1, 1 ] ] gap> UpdateMap( he.irreducibles[2], fus, last[1] + s.irreducibles[1] ); gap> fus; [ 1, 2, 2, 3, 4, 4, 8, 7, 9, 9, 9, 10, 10, 18, 18, 25, 25, [ 26, 27 ], [ 26, 27 ], 2, [ 6, 7 ], [ 6, 7 ], [ 6, 7 ], 10, 10, 17, 17, 18, [ 19, 20 ], [ 19, 20 ] ]| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{CommutativeDiagram} 'CommutativeDiagram( , , , )'\\ 'CommutativeDiagram( , , , , )' If \[ 'CompositionMaps(, ) = CompositionMaps(, )' \] shall hold, the consistency is checked and the four maps will be improved according to this condition. If a record with fields 'imp1', 'imp2', 'imp3', 'imp4' (all lists) is entered as parameter, only diagrams containing elements of 'imp' as positions in the -th paramap are considered. 'CommutativeDiagram' returns 'false' if an inconsistency was found, otherwise a record is returned that contains four lists 'imp1', \ldots, 'imp4', where 'imp' is the list of classes where the -th paramap was improved. | gap> map1:= [ [ 1, 2, 3 ], [ 1, 3 ] ];; gap> map2:= [ [ 1, 2 ], 1, [ 1, 3 ] ];; gap> map3:= [ [ 2, 3 ], 3 ];; map4:= [ , 1, 2, [ 1, 2 ] ];; gap> CommutativeDiagram( map1, map2, map3, map4 ); rec( imp1 := [ 2 ], imp2 := [ 1 ], imp3 := [ ], imp4 := [ ] ) gap> imp:= last;; map1; map2; map3; map4; [ [ 1, 2, 3 ], 1 ] [ 2, 1, [ 1, 3 ] ] [ [ 2, 3 ], 3 ] [ , 1, 2, [ 1, 2 ] ] gap> CommutativeDiagram( map1, map2, map3, map4, imp ); rec( imp1 := [ ], imp2 := [ ], imp3 := [ ], imp4 := [ ] )| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{TransferDiagram} 'TransferDiagram( , , )'\\ 'TransferDiagram( , , , )' Like in "CommutativeDiagram", it is checked that \[ 'CompositionMaps( , ) = CompositionMaps( , )' \] holds for the paramaps , and , that means the paramap occurs twice in each commutative diagram. Additionally, "CheckFixedPoints" 'CheckFixedPoints' is called. If a record with fields 'impinside1', 'impbetween' and 'impinside2' is specified, only those diagrams with elements of 'impinside1' as positions in , elements of 'impbetween' as positions in or elements of 'impinside2' as positions in are considered. When an inconsistency occurs, the program immediately returns 'false'; else it returns a record with lists 'impinside1', 'impbetween' and 'impinside2' of found improvements. | gap> s:= CharTable( "2F4(2)" );; ru:= CharTable( "Ru" );; gap> fus:= InitFusion( s, ru );; gap> permchar:= Sum( Sublist( ru.irreducibles, [ 1, 5, 6 ] ) );; gap> CheckPermChar( s, ru, fus, permchar );; fus; [ 1, 2, 2, 4, 5, 7, 8, 9, 11, 14, 14, [ 13, 15 ], 16, [ 18, 19 ], 20, [ 25, 26 ], [ 25, 26 ], 5, 5, 6, 8, 14, [ 13, 15 ], [ 18, 19 ], [ 18, 19 ], [ 25, 26 ], [ 25, 26 ], 27, 27 ] gap> TransferDiagram( s.powermap[2], fus, ru.powermap[2] ); rec( impinside1 := [ ], impbetween := [ 12, 23 ], impinside2 := [ ] ) gap> TransferDiagram( s.powermap[3], fus, ru.powermap[3] ); rec( impinside1 := [ ], impbetween := [ 14, 24, 25 ], impinside2 := [ ] ) gap> TransferDiagram( s.powermap[2], fus, ru.powermap[2], last ); rec( impinside1 := [ ], impbetween := [ ], impinside2 := [ ] ) gap> fus; [ 1, 2, 2, 4, 5, 7, 8, 9, 11, 14, 14, 15, 16, 18, 20, [ 25, 26 ], [ 25, 26 ], 5, 5, 6, 8, 14, 13, 19, 19, [ 25, 26 ], [ 25, 26 ], 27, 27 ]| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Indeterminateness}\index{map!indeterminateness} 'Indeterminateness( )' returns the indeterminateness of , i.e. the number of maps contained in the parametrized map | gap> m11:= CharTable( "M11" );; m12:= CharTable( "M12" );; gap> fus:= InitFusion( m11, m12 ); [ 1, [ 2, 3 ], [ 4, 5 ], [ 6, 7 ], 8, [ 9, 10 ], [ 11, 12 ], [ 11, 12 ], [ 14, 15 ], [ 14, 15 ] ] gap> Indeterminateness( fus ); 256 gap> TestConsistencyMaps( m11.powermap, fus, m12.powermap );; fus; [ 1, 3, 4, [ 6, 7 ], 8, 10, [ 11, 12 ], [ 11, 12 ], [ 14, 15 ], [ 14, 15 ] ] gap> Indeterminateness( fus ); 32| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{PrintAmbiguity} 'PrintAmbiguity( , )' prints for each character in its position, its indeterminateness with respect to and the list of ambiguous classes | gap> s:= CharTable( "2F4(2)" );; ru:= CharTable( "Ru" );; gap> fus:= InitFusion( s, ru );; gap> permchar:= Sum( Sublist( ru.irreducibles, [ 1, 5, 6 ] ) );; gap> CheckPermChar( s, ru, fus, permchar );; fus; [ 1, 2, 2, 4, 5, 7, 8, 9, 11, 14, 14, [ 13, 15 ], 16, [ 18, 19 ], 20, [ 25, 26 ], [ 25, 26 ], 5, 5, 6, 8, 14, [ 13, 15 ], [ 18, 19 ], [ 18, 19 ], [ 25, 26 ], [ 25, 26 ], 27, 27 ] gap> PrintAmbiguity( Sublist( ru.irreducibles, [ 1 .. 8 ] ), fus ); 1 1 [ ] 2 16 [ 16, 17, 26, 27 ] 3 16 [ 16, 17, 26, 27 ] 4 32 [ 12, 14, 23, 24, 25 ] 5 4 [ 12, 23 ] 6 1 [ ] 7 32 [ 12, 14, 23, 24, 25 ] 8 1 [ ] gap> Indeterminateness( fus ); 512| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Powermap}\index{powermaps} 'Powermap( , )'\\ 'Powermap( , , )' returns a list of possibilities for the -th powermap of the character table . The optional record may have the following fields\: 'chars':\\ a list of characters which are used for the check of kernels (see "ConsiderKernels"), the test of congruences (see "Congruences") and the test of scalar products of symmetrisations (see "PowermapsAllowedBySymmetrisations"); the default is '.irreducibles' 'powermap':\\ a (parametrized) map which is an approximation of the desired map 'decompose':\\ a boolean; if 'true', the symmetrisations of 'chars' must have all constituents in 'chars', that will be used in the algorithm; if 'chars' is not bound and '.irreducibles' is complete, the default value of 'decompose' is 'true', otherwise 'false' 'quick':\\ a boolean; if 'true', the subroutines are called with the option '\"quick\"'; especially, a unique map will be returned immediately without checking all symmetrisations; the default value is 'false' 'parameters':\\ a record with fields 'maxamb', 'minamb' and 'maxlen' which control the subroutine "PowermapsAllowedBySymmetrisations"\:\ It only uses characters with actual indeterminateness up to 'maxamb', tests decomposability only for characters with actual indeterminateness at least 'minamb' and admits a branch only according to a character if there is one with at most 'maxlen' possible minus-characters. | # cf.\ example in "InitPowermap" gap> t:= CharTable( "U4(3).4" );; gap> pow:= Powermap( t, 2 ); [ [ 1, 1, 3, 4, 5, 2, 2, 8, 3, 4, 11, 12, 6, 14, 9, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 9, 10, 11, 12, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 20, 20, 20, 20, 22, 22, 24, 24, 25, 26, 28, 28, 29, 29 ] ]| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{SubgroupFusions}\index{fusions}\index{subgroup fusions} 'SubgroupFusions( , )'\\ 'SubgroupFusions( , , )' returns the list of all subgroup fusion maps from into . The optional record may have the following fields\: 'chars':\\ a list of characters of which will be restricted to , (see "FusionsAllowedByRestrictions"); the default is '' 'subchars':\\ a list of characters of which are constituents of the retrictions of 'chars', the default is '.irreducibles' 'fusionmap':\\ a (parametrized) map which is an approximation of the desired map 'decompose':\\ a boolean; if 'true', the restrictions of 'chars' must have all constituents in 'subchars', that will be used in the algorithm; if 'subchars' is not bound and '.irreducibles' is complete, the default value of 'decompose' is 'true', otherwise 'false' 'permchar':\\ a permutation character; only those fusions are computed which afford that permutation character (see "CheckPermChar") 'quick':\\ a boolean; if 'true', the subroutines are called with the option '\"quick\"'; especially, a unique map will be returned immediately without checking all symmetrisations; the default value is 'false' 'parameters':\\ a record with fields 'maxamb', 'minamb' and 'maxlen' which control the subroutine "FusionsAllowedByRestrictions"\:\ It only uses characters with actual indeterminateness up to 'maxamb', tests decomposability only for characters with actual indeterminateness at least 'minamb' and admits a branch only according to a character if there is one with at most 'maxlen' possible restrictions. | # cf.\ example in "FusionsAllowedByRestrictions" gap> s:= CharTable( "U3(3)" );; t:= CharTable( "J4" );; gap> SubgroupFusions( s, t, rec( quick:= true ) ); [ [ 1, 2, 4, 4, 5, 5, 6, 10, 12, 13, 14, 14, 21, 21 ], [ 1, 2, 4, 4, 5, 5, 6, 10, 13, 12, 14, 14, 21, 21 ], [ 1, 2, 4, 4, 6, 6, 6, 10, 12, 13, 15, 15, 22, 22 ], [ 1, 2, 4, 4, 6, 6, 6, 10, 12, 13, 16, 16, 22, 22 ], [ 1, 2, 4, 4, 6, 6, 6, 10, 13, 12, 15, 15, 22, 22 ], [ 1, 2, 4, 4, 6, 6, 6, 10, 13, 12, 16, 16, 22, 22 ] ]| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{InitPowermap}\index{powermaps} 'InitPowermap( , )' computes a (probably parametrized, see "More about Maps and Parametrized Maps") first approximation of of the -th powermap of the character table , using that for any class 'i' of , the following properties hold\: The centralizer order of the image is a multiple of the centralizer order of 'i'. If the element order of 'i' is relative prime to , the centralizer orders of 'i' and its image must be equal. If divides the element order of the class 'i', the element order of its image must be $ / $; otherwise the element orders of 'i' and its image must be equal. Of course, this is used only if the element orders are stored on the table. If no -th powermap is possible because of these properties, 'false' is returned. Otherwise 'InitPowermap' returns the parametrized map. | # cf.\ example in "Powermap" gap> t:= CharTable( "U4(3).4" );; gap> pow:= InitPowermap( t, 2 ); [ 1, 1, 3, 4, 5, [ 2, 16 ], [ 2, 16, 17 ], 8, 3, [ 3, 4 ], [ 11, 12 ], [ 11, 12 ], [ 6, 7, 18, 19, 30, 31, 32, 33 ], 14, [ 9, 20 ], 1, 1, 2, 2, 3, [ 3, 4, 5 ], [ 3, 4, 5 ], [ 6, 7, 18, 19, 30, 31, 32, 33 ], 8, 9, 9, [ 9, 10, 20, 21, 22 ], [ 11, 12 ], [ 11, 12 ], 16, 16, [ 2, 16 ], [ 2, 16 ], 17, 17, [ 6, 18, 30, 31, 32, 33 ], [ 6, 18, 30, 31, 32, 33 ], [ 6, 7, 18, 19, 30, 31, 32, 33 ], [ 6, 7, 18, 19, 30, 31, 32, 33 ], 20, 20, [ 9, 20 ], [ 9, 20 ], [ 9, 10, 20, 21, 22 ], [ 9, 10, 20, 21, 22 ], 24, 24, [ 15, 25, 26, 40, 41, 42, 43 ], [ 15, 25, 26, 40, 41, 42, 43 ], [ 28, 29 ], [ 28, 29 ], [ 28, 29 ], [ 28, 29 ] ] # continued in "Congruences"| 'InitPowermap' is used by "Powermap" 'Powermap'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Congruences}\index{powermaps} 'Congruences( , , , )'\\ 'Congruences( , , , , \"quick\" )' improves the parametrized map (see "More about Maps and Parametrized Maps") that is an approximation of the -th powermap of the character table \: For $G$ a group with character table , $g \in G$ and a character $\chi$ of , the congruence \['GaloisCyc( \chi(), )' \equiv\chi(g^{})\pmod{}\] holds; if the representative order of $g$ is relative prime to , we have \['GaloisCyc( \chi(g), ) = \chi(g^{})'.\] 'Congruences' checks these congruences for the (virtual ) characters in the list . If '\"quick\"' is specified, only those classes are considered for which is ambiguous. If there are classes for which no image is possible, 'false' is returned, otherwise 'Congruences' returns 'true'. | # see example in "InitPowermap" gap> Congruences( t, t.irreducibles, pow, 2 ); pow; true [ 1, 1, 3, 4, 5, [ 2, 16 ], [ 2, 16, 17 ], 8, 3, 4, 11, 12, [ 6, 7, 18, 19 ], 14, [ 9, 20 ], 1, 1, 2, 2, 3, 4, 5, [ 6, 7, 18, 19 ], 8, 9, 9, [ 10, 21 ], 11, 12, 16, 16, [ 2, 16 ], [ 2, 16 ], 17, 17, [ 6, 18 ], [ 6, 18 ], [ 6, 7, 18, 19 ], [ 6, 7, 18, 19 ], 20, 20, [ 9, 20 ], [ 9, 20 ], 22, 22, 24, 24, [ 15, 25, 26 ], [ 15, 25, 26 ], 28, 28, 29, 29 ] # continued in "ConsiderKernels"| 'Congruences' is used by "Powermap" 'Powermap'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{ConsiderKernels}\index{powermaps} 'ConsiderKernels( , , , )'\\ 'ConsiderKernels( , , , , \"quick\" )' improves the parametrized map (see "More about Maps and Parametrized Maps") that is an approximation of the -th powermap of the character table \: For $G$ a group with character table , the kernel of each character in the list is a normal subgroup of $G$, so for every $g \in 'Kernel( )'$ we have $g^{} \in 'Kernel( )'$. Depending on the order of the factor group modulo 'Kernel( )', there are two further properties\: If the order is relative prime to , for each $g \notin 'Kernel( )'$ the -th power is not contained in 'Kernel( )'; if the order is equal to , the -th powers of all elements lie in 'Kernel( )'. If '\"quick\"' is specified, only those classes are considered for which is ambiguous. If 'Kernel( )' has an order not dividing '.order' for an element of , or if no image is possible for a class, 'false' is returned; otherwise 'ConsiderKernels' returns 'true'. *Note* that must consist of ordinary characters, since the kernel of a virtual character is not defined. | # see example in "Congruences" gap> ConsiderKernels( t, t.irreducibles, pow, 2 ); pow; true [ 1, 1, 3, 4, 5, 2, 2, 8, 3, 4, 11, 12, [ 6, 7 ], 14, 9, 1, 1, 2, 2, 3, 4, 5, [ 6, 7, 18, 19 ], 8, 9, 9, [ 10, 21 ], 11, 12, 16, 16, [ 2, 16 ], [ 2, 16 ], 17, 17, [ 6, 18 ], [ 6, 18 ], [ 6, 7, 18, 19 ], [ 6, 7, 18, 19 ], 20, 20, [ 9, 20 ], [ 9, 20 ], 22, 22, 24, 24, [ 15, 25, 26 ], [ 15, 25, 26 ], 28, 28, 29, 29 ] # continued in "PowermapsAllowedBySymmetrisations"| 'ConsiderKernels' is used by "Powermap" 'Powermap'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{ConsiderSmallerPowermaps}\index{powermaps} 'ConsiderSmallerPowermaps( , , )'\\ 'ConsiderSmallerPowermaps( , , , \"quick\" )' improves the parametrized map (see chapter "Maps and Parametrized Maps") that is an approximation of the -th powermap of the character table \: If $ > '.orders[i]'$ for a class 'i', try to improve at class 'i' using that for $g$ in class 'i', $g_i^{} = g_i^{\ 'mod\ .orders[i]'}$ holds; \\ so if the '( mod .orders[i])'-th powermap at class 'i' is determined by the maps stored in '.powermap', this information is used. If '\"quick\"' is specified, only those classes are considered for which is ambiguous. If there are classes for which no image is possible, 'false' is returned, otherwise 'true'. *Note*\:\ If '.orders' is unbound, 'true' is returned without tests. | gap> t:= CharTable( "3.A6" );; init:= InitPowermap( t, 5 );; gap> Indeterminateness( init ); 4096 gap> ConsiderSmallerPowermaps( t, init, 5 );; gap> Indeterminateness( init ); 256 | 'ConsiderSmallerPowermaps' is used by "Powermap" 'Powermap'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{InitFusion}\index{subgroup fusions} 'InitFusion( , )' computes a (probably parametrized, see "More about Maps and Parametrized Maps") first approximation of of the subgroup fusion from the character table into the character table , using that for any class 'i' of , the centralizer order of the image is a multiple of the centralizer order of 'i' and the element order of 'i' is equal to the element order of its image (used only if element orders are stored on the tables). If no fusion map is possible because of these properties, 'false' is returned. Otherwise 'InitFusion' returns the parametrized map. | gap> s:= CharTable( "2F4(2)" );; ru:= CharTable( "Ru" );; gap> fus:= InitFusion( s, ru ); [ 1, 2, 2, 4, [ 5, 6 ], [ 5, 6, 7, 8 ], [ 5, 6, 7, 8 ], [ 9, 10 ], 11, 14, 14, [ 13, 14, 15 ], [ 16, 17 ], [ 18, 19 ], 20, [ 25, 26 ], [ 25, 26 ], [ 5, 6 ], [ 5, 6 ], [ 5, 6 ], [ 5, 6, 7, 8 ], [ 13, 14, 15 ], [ 13, 14, 15 ], [ 18, 19 ], [ 18, 19 ], [ 25, 26 ], [ 25, 26 ], [ 27, 28, 29 ], [ 27, 28, 29 ] ]| 'InitFusion' is used by "SubgroupFusions" 'SubgroupFusions'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{CheckPermChar} \index{subgroup fusions} \index{permutation character} 'CheckPermChar( , , , )' tries to improve the parametrized fusion (see Chapter "Maps and Parametrized Maps") from the character table into the character table using the permutation character that belongs to the required fusion\:\ A possible image 'x' of class 'i' is excluded if class 'i' is too large, and a possible image 'y' of class 'i' is the right image if 'y' must be the image of {\em all} classes where 'y' is a possible image. 'CheckPermChar' returns 'true' if no inconsistency occurred, and 'false' otherwise. | gap> fus:= InitFusion( s, ru );; # cf. example in "InitFusion" gap> permchar:= Sum( Sublist( ru.irreducibles, [ 1, 5, 6 ] ) );; gap> CheckPermChar( s, ru, fus, permchar );; fus; [ 1, 2, 2, 4, 5, 7, 8, 9, 11, 14, 14, [ 13, 15 ], 16, [ 18, 19 ], 20, [ 25, 26 ], [ 25, 26 ], 5, 5, 6, 8, 14, [ 13, 15 ], [ 18, 19 ], [ 18, 19 ], [ 25, 26 ], [ 25, 26 ], 27, 27 ]| 'CheckPermChar' is used by "SubgroupFusions" 'SubgroupFusions'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{CheckFixedPoints}\index{subgroup fusions} 'CheckFixedPoints( , , )' If the parametrized map (see "More about Maps and Parametrized Maps") transfers the parametrized map to , i.e.\ $ \circ = \circ $, must map fixed points of to fixed points of . Using this property, 'CheckFixedPoints' tries to improve and . If an inconsistency occurs, 'false' is returned. Otherwise, 'CheckFixedPoints' returns the list of classes where improvements were found. | gap> s:= CharTable( "L4(3).2_2" );; o7:= CharTable( "O7(3)" );; gap> fus:= InitFusion( s, o7 );; gap> CheckFixedPoints( s.powermap[5], fus, o7.powermap[5] ); [ 48, 49 ] gap> fus:= InitFusion( s, o7 );; Sublist( fus, [ 48, 49 ] ); [ [ 54, 55, 56, 57 ], [ 54, 55, 56, 57 ] ] gap> CheckFixedPoints( s.powermap[5], fus, o7.powermap[5] ); [ 48, 49 ] gap> Sublist( fus, [ 48, 49 ] ); [ [ 56, 57 ], [ 56, 57 ] ]| 'CheckFixedPoints' is used by "SubgroupFusions" 'SubgroupFusions'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{TestConsistencyMaps}\index{subgroup fusions} 'TestConsistencyMaps( , , )'\\ 'TestConsistencyMaps( , , , )' Like in "TransferDiagram", it is checked that parametrized maps (see chapter "Maps and Parametrized Maps") commute\: For all positions 'i' where both '[i]' and '[i]' are bound, \[ 'CompositionMaps( , [i] ) = CompositionMaps( [i], )' \] shall hold, so occurs in diagrams for all considered elements of resp. , and it occurs twice in each diagram. If a set is specified, only those diagrams with elements of as preimages of are considered. 'TestConsistencyMaps' stores all found improvements in and elements of and . When an inconsistency occurs, the program immediately returns 'false'; otherwise 'true' is returned. 'TestConsistencyMaps' stops if no more improvements of are possible. E.g.\ if was unique from the beginning, the powermaps will not be improved. To transfer powermaps by fusions, use "TransferDiagram" 'TransferDiagram'. | gap> s:= CharTable( "2F4(2)" );; ru:= CharTable( "Ru" );; gap> fus:= InitFusion( s, ru );; gap> permchar:= Sum( Sublist( ru.irreducibles, [ 1, 5, 6 ] ) );; gap> CheckPermChar( s, ru, fus, permchar );; fus; [ 1, 2, 2, 4, 5, 7, 8, 9, 11, 14, 14, [ 13, 15 ], 16, [ 18, 19 ], 20, [ 25, 26 ], [ 25, 26 ], 5, 5, 6, 8, 14, [ 13, 15 ], [ 18, 19 ], [ 18, 19 ], [ 25, 26 ], [ 25, 26 ], 27, 27 ] gap> TestConsistencyMaps( s.powermap, fus, ru.powermap ); true gap> fus; [ 1, 2, 2, 4, 5, 7, 8, 9, 11, 14, 14, 15, 16, 18, 20, [ 25, 26 ], [ 25, 26 ], 5, 5, 6, 8, 14, 13, 19, 19, [ 25, 26 ], [ 25, 26 ], 27, 27 ] gap> Indeterminateness( fus ); 16| 'TestConsistencyMaps' is used by "SubgroupFusions" 'SubgroupFusions'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{ConsiderTableAutomorphisms}\index{subgroup fusions} \index{table automorphisms} 'ConsiderTableAutomorphisms( , )' improves the parametrized subgroup fusion map (see "More about Maps and Parametrized Maps")\:\ Let $T$ be the permutation group that has the list as generators, let $T_0$ be the subgroup of $T$ that is maximal with the property that $T_0$ operates on the set of fusions contained in by permutation of images. 'ConsiderTableAutomorphisms' replaces orbits by representatives at suitable positions so that afterwards exactly one representative of fusion maps (that is contained in ) in every orbit under the operation of $T_0$ is contained in . The list of positions where improvements were found is returned. | gap> fus:= InitFusion( s, ru );; gap> permchar:= Sum( Sublist( ru.irreducibles, [ 1, 5, 6 ] ) );; gap> CheckPermChar( s, ru, fus, permchar );; fus; [ 1, 2, 2, 4, 5, 7, 8, 9, 11, 14, 14, [ 13, 15 ], 16, [ 18, 19 ], 20, [ 25, 26 ], [ 25, 26 ], 5, 5, 6, 8, 14, [ 13, 15 ], [ 18, 19 ], [ 18, 19 ], [ 25, 26 ], [ 25, 26 ], 27, 27 ] gap> ConsiderTableAutomorphisms( fus, ru.automorphisms ); [ 16 ] gap> fus; [ 1, 2, 2, 4, 5, 7, 8, 9, 11, 14, 14, [ 13, 15 ], 16, [ 18, 19 ], 20, 25, [ 25, 26 ], 5, 5, 6, 8, 14, [ 13, 15 ], [ 18, 19 ], [ 18, 19 ], [ 25, 26 ], [ 25, 26 ], 27, 27 ]| 'ConsiderTableAutomorphisms' is used by 'SubgroupFusions' (see "SubgroupFusions"). Note that the function 'SubgroupFusions' forms orbits of fusion maps under table automorphisms, but it returns all possible fusions. If you want to get only orbit representatives, use the function 'RepresentativesFusions' (see "RepresentativesFusions"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{PowermapsAllowedBySymmetrisations}\index{powermaps} 'PowermapsAllowedBySymmetrisations( , , , ,'\\ | |', )' returns a list of (possibly parametrized, see "More about Maps and Parametrized Maps") maps which are contained in the parametrized map and which have the property that for all $\chi$ in the list of characters of the character table , the symmetrizations \[ \chi^{p-} = |M|'inusCharacter( '\chi', , )' \] (see "MinusCharacter") have nonnegative integral scalar products with all characters in the list . must be a record with fields 'maxlen':\\ an integer that controls the position where branches take place 'contained':\\ a function, usually "ContainedCharacters" or "ContainedPossibleCharacters"; for a symmetrization , it returns the list 'contained( , , )' 'minamb', 'maxamb':\\ two arbitrary objects; is called only for symmetrizations with \[ 'minamb \< Indeterminateness( ) \< maxamb' \] 'quick':\\ a boolean; if it is true, the scalar products of uniquely determined symmetrizations are not checked. will be improved, i.e.\ is changed by the algorithm. If there is no character left which allows an immediate improvement but there are characters in with indeterminateness of the symmetrizations bigger than '.minamb', a branch is necessary. Two kinds of branches may occur\:\ If '.contained( , , )' has length at most '.maxlen', the union of maps allowed by the characters in is computed; otherwise a suitable class 'c' is taken which is significant for some character, and the union of all admissible maps with image 'x' on 'c' is computed, where 'x' runs over '[c]'. | # see example in "ConsiderKernels" gap> t := CharTable( "U4(3).4" );; gap> PowermapsAllowedBySymmetrisations(t,t.irreducibles,t.irreducibles, > pow, 2, rec( maxlen:=10, contained:=ContainedPossibleCharacters, > minamb:= 2, maxamb:= "infinity", quick:= false ) ); [ [ 1, 1, 3, 4, 5, 2, 2, 8, 3, 4, 11, 12, 6, 14, 9, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 9, 10, 11, 12, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 20, 20, 20, 20, 22, 22, 24, 24, 25, 26, 28, 28, 29, 29 ] ] gap> t.powermap[2] = last[1]; true| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{FusionsAllowedByRestrictions}\index{subgroup fusions} 'FusionsAllowedByRestrictions( , , , ,'\\ | |', )' returns a list of (possibly parametrized, see "More about Maps and Parametrized Maps") maps which are contained in the parametrized map and which have the property that for all $\chi$ in the list of characters of the character table , the restrictions \[ \chi_{} = 'CompositionMaps( '\chi', )' \] (see "CompositionMaps") have nonnegative integral scalar products with all characters in the list . must be a record with fields 'maxlen':\\ an integer that controls the position where branches take place 'contained':\\ a function, usually "ContainedCharacters" or "ContainedPossibleCharacters"; for a restriction , it returns the list 'contained( , , )'; 'minamb', 'maxamb':\\ two arbitrary objects; is called only for restrictions with 'minamb \< Indeterminateness( ) \< maxamb'; 'quick':\\ a boolean value; if it is true, the scalar products of uniquely determined restrictions are not checked. will be improved, i.e.\ is changed by the algorithm. If there is no character left which allows an immediate improvement but there are characters in with indeterminateness of the restrictions bigger than '.minamb', a branch is necessary. Two kinds of branches may occur\:\ If '.contained( , , )' has length at most '.maxlen', the union of maps allowed by the characters in is computed; otherwise a suitable class 'c' is taken which is significant for some character, and the union of all admissible maps with image 'x' on 'c' is computed, where 'x' runs over '[c]'. | gap> s:= CharTable( "U3(3)" );; t:= CharTable( "J4" );; gap> fus:= InitFusion( s, t );; gap> TestConsistencyMaps( s.powermap, fus, t.powermap );; gap> ConsiderTableAutomorphisms( fus, t.automorphisms );; fus; [ 1, 2, 4, 4, [ 5, 6 ], [ 5, 6 ], [ 5, 6 ], 10, 12, [ 12, 13 ], [ 14, 15, 16 ], [ 14, 15, 16 ], [ 21, 22 ], [ 21, 22 ] ] gap> FusionsAllowedByRestrictions( s, t, s.irreducibles, > t.irreducibles, fus, rec( maxlen:= 10, > contained:= ContainedPossibleCharacters, > minamb:= 2, maxamb:= "infinity", quick:= false ) ); [ [ 1, 2, 4, 4, 5, 5, 6, 10, 12, 13, 14, 14, 21, 21 ], [ 1, 2, 4, 4, 6, 6, 6, 10, 12, 13, 15, 15, 22, 22 ], [ 1, 2, 4, 4, 6, 6, 6, 10, 12, 13, 16, 16, 22, 22 ] ] # cf.\ example in "SubgroupFusions"| 'FusionsAllowedByRestrictions' is used by "SubgroupFusions" 'SubgroupFusions'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{OrbitFusions}\index{subgroup fusions} \index{table automorphisms} 'OrbitFusions( , , )' returns the orbit of the subgroup fusion map under the operations of maximal admissible subgroups of the table automorphism groups of the character tables. is a list of generators of the automorphisms of the subgroup table, is a list of generators of the automorphisms of the supergroup table. | gap> s:= CharTable( "U3(3)" );; t:= CharTable( "J4" );; gap> GetFusionMap( s, t ); [ 1, 2, 4, 4, 5, 5, 6, 10, 12, 13, 14, 14, 21, 21 ] gap> OrbitFusions( s.automorphisms, last, t.automorphisms ); [ [ 1, 2, 4, 4, 5, 5, 6, 10, 12, 13, 14, 14, 21, 21 ], [ 1, 2, 4, 4, 5, 5, 6, 10, 13, 12, 14, 14, 21, 21 ] ]| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{OrbitPowermaps}\index{powermaps} \index{matrix automorphisms} 'OrbitPowermaps( , )' returns the orbit of the powermap under the operation of the subgroup of the maximal admissible subgroup of the matrix automorphisms of the corresponding character table. | gap> t:= CharTable( "3.McL" );; gap> maut:= MatAutomorphisms( t.irreducibles, [], Group( () ) ); Group( (55,58)(56,59)(57,60)(61,64)(62,65)(63,66), (35,36), (26,29) (27,30)(28,31)(49,52)(50,53)(51,54), (40,43)(41,44)(42,45), ( 2, 3) ( 5, 6)( 8, 9)(12,13)(15,16)(18,19)(21,22)(24,25)(27,28)(30,31)(33,34) (38,39)(41,42)(44,45)(47,48)(50,51)(53,54)(56,57)(59,60)(62,63) (65,66) ) gap> OrbitPowermaps( t.powermap[3], maut ); [ [ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17, 4, 4, 4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 8, 9, 37, 37, 37, 40, 40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49, 49, 14, 14, 14, 14, 14, 14, 37, 37, 37, 37, 37, 37 ], [ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17, 4, 4, 4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 9, 8, 37, 37, 37, 40, 40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49, 49, 14, 14, 14, 14, 14, 14, 37, 37, 37, 37, 37, 37 ] ]| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{RepresentativesFusions}\index{subgroup fusions} \index{table automorphisms} 'RepresentativesFusions( , , )'\\ 'RepresentativesFusions( , , )' returns a list of representatives of the list of subgroup fusion maps under the operations of maximal admissible subgroups of the table automorphism groups of the character tables. is a list of generators of the automorphisms of the subgroup table, is a list of generators of the automorphisms of the supergroup table. if the parameters and (character tables) are used, the values of '.automorphisms' and '.automorphisms' will be taken. | gap> s:= CharTable( "2F4(2)" );; ru:= CharTable( "Ru" );; gap> SubgroupFusions( s, ru ); [ [ 1, 2, 2, 4, 5, 7, 8, 9, 11, 14, 14, 15, 16, 18, 20, 25, 26, 5, 5, 6, 8, 14, 13, 19, 19, 26, 25, 27, 27 ], [ 1, 2, 2, 4, 5, 7, 8, 9, 11, 14, 14, 15, 16, 18, 20, 26, 25, 5, 5, 6, 8, 14, 13, 19, 19, 25, 26, 27, 27 ] ] gap> RepresentativesFusions( s, last, ru ); [ [ 1, 2, 2, 4, 5, 7, 8, 9, 11, 14, 14, 15, 16, 18, 20, 25, 26, 5, 5, 6, 8, 14, 13, 19, 19, 26, 25, 27, 27 ] ]| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{RepresentativesPowermaps}\index{powermaps}% \index{matrix automorphisms} 'RepresentativesPowermaps( , )' returns a list of representatives of the list of powermaps under the operation of a subgroup of the maximal admissible subgroup of matrix automorphisms of irreducible characters of the corresponding character table. | gap> t:= CharTable( "3.McL" );; gap> maut:= MatAutomorphisms( t.irreducibles, [], Group( () ) ); Group( (55,58)(56,59)(57,60)(61,64)(62,65)(63,66), (35,36), (26,29) (27,30)(28,31)(49,52)(50,53)(51,54), (40,43)(41,44)(42,45), ( 2, 3) ( 5, 6)( 8, 9)(12,13)(15,16)(18,19)(21,22)(24,25)(27,28)(30,31)(33,34) (38,39)(41,42)(44,45)(47,48)(50,51)(53,54)(56,57)(59,60)(62,63) (65,66) ) gap> Powermap( t, 3 ); [ [ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17, 4, 4, 4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 9, 8, 37, 37, 37, 40, 40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49, 49, 14, 14, 14, 14, 14, 14, 37, 37, 37, 37, 37, 37 ], [ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17, 4, 4, 4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 8, 9, 37, 37, 37, 40, 40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49, 49, 14, 14, 14, 14, 14, 14, 37, 37, 37, 37, 37, 37 ] ] gap> RepresentativesPowermaps( last, maut ); [ [ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17, 4, 4, 4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 8, 9, 37, 37, 37, 40, 40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49, 49, 14, 14, 14, 14, 14, 14, 37, 37, 37, 37, 37, 37 ] ]| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Indirected}\index{map!indirection by a} 'Indirected( , )' We have \[ 'Indirected( , )[i]' = '[ [i] ]', \] if this value is unique; otherwise it is set unknown (see chapter "Unknowns"). (For a parametrized indirection, see "CompositionMaps".) | gap> m12:= CharTable( "M12" );; gap> fus:= [ 1, 3, 4, [ 6, 7 ], 8, 10, [ 11, 12 ], [ 11, 12 ], > [ 14, 15 ], [ 14, 15 ] ];; # parametrized subgroup fusion # from $M_{11}$ gap> chars:= Sublist( m12.irreducibles, [ 1 .. 6 ] );; gap> List( chars, x -> Indirected( x, fus ) ); [ [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ], [ 11, 3, 2, Unknown(1), 1, 0, Unknown(2), Unknown(3), 0, 0 ], [ 11, 3, 2, Unknown(4), 1, 0, Unknown(5), Unknown(6), 0, 0 ], [ 16, 0, -2, 0, 1, 0, 0, 0, Unknown(7), Unknown(8) ], [ 16, 0, -2, 0, 1, 0, 0, 0, Unknown(9), Unknown(10) ], [ 45, -3, 0, 1, 0, 0, -1, -1, 1, 1 ] ]| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Powmap} 'Powmap( , )'\\ 'Powmap( , , )' The first form returns the -th powermap where is the powermap of a character table (see "Character Table Records"). If the -th position in is bound, this map is returned, otherwise it is computed from the (necessarily stored) powermaps of the prime divisors of . The second form returns the image of under the -th powermap; for any valid class , we have 'Powmap( , )[ ] = Powmap( , , )'. The entries of may be parametrized maps (see "More about Maps and Parametrized Maps"). | gap> t:= CharTable( "3.McL" );; gap> Powmap( t.powermap, 3 ); [ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17, 4, 4, 4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 8, 9, 37, 37, 37, 40, 40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49, 49, 14, 14, 14, 14, 14, 14, 37, 37, 37, 37, 37, 37 ] gap> Powmap( t.powermap, 27 ); [ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17, 4, 4, 4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 1, 1, 37, 37, 37, 40, 40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49, 49, 14, 14, 14, 14, 14, 14, 37, 37, 37, 37, 37, 37 ] gap> Lcm( t.orders ); Powmap( t.powermap, last ); 27720 [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{ElementOrdersPowermap} 'ElementOrdersPowermap( )' returns the list of element orders given by the maps in the powermap . The entries at positions where the powermaps do not uniquely determine the element order are set to unknowns (see chapter "Unknowns"). | gap> t:= CharTable( "3.J3.2" );; t.powermap; [ , [ 1, 2, 1, 2, 5, 6, 7, 3, 4, 10, 11, 12, 5, 6, 8, 9, 18, 19, 17, 10, 11, 12, 13, 14, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 1, 3, 7, 8, 8, 13, 18, 19, 17, 23, 23, 28, 30 ], [ 1, 1, 3, 3, 1, 1, 1, 8, 8, 10, 10, 10, 3, 3, 15, 15, 7, 7, 7, 20, 20, 20, 8, 8, 10, 10, 10, 30, 30, 28, 28, 32, 32, 32, 35, 36, 35, 38, 39, 36, 37, 37, 37, 38, 38, 47, 46 ],, [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 2, 13, 14, 15, 16, 19, 17, 18, 3, 4, 4, 23, 24, 5, 6, 6, 30, 31, 28, 29, 32, 34, 33, 35, 36, 37, 38, 39, 40, 43, 41, 42, 44, 45, 47, 46 ],,,,,,,,,,,, [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 1, 2, 1, 2, 32, 34, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 44, 35, 35 ],, [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 1, 2, 2, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47 ] ] gap> ElementOrdersPowermap( last ); [ 1, 3, 2, 6, 3, 3, 3, 4, 12, 5, 15, 15, 6, 6, 8, 24, 9, 9, 9, 10, 30, 30, 12, 12, 15, 15, 15, 17, 51, 17, 51, 19, 57, 57, 2, 4, 6, 8, 8, 12, 18, 18, 18, 24, 24, 34, 34 ] gap> Unbind( t.powermap[17] ); ElementOrdersPowermap( t.powermap ); [ 1, 3, 2, 6, 3, 3, 3, 4, 12, 5, 15, 15, 6, 6, 8, 24, 9, 9, 9, 10, 30, 30, 12, 12, 15, 15, 15, Unknown(11), Unknown(12), Unknown(13), Unknown(14), 19, 57, 57, 2, 4, 6, 8, 8, 12, 18, 18, 18, 24, 24, Unknown(15), Unknown(16) ]|