Besides the characters, power maps are an important part of a character table, see Section 73.1. Often their computation is not easy, and if the table has no access to the underlying group then 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 a group and a subgroup or of a group and a factor group are used, also class fusion maps (see Section 73.3) must be known to get information about the embedding or simply to induce or restrict characters, see Section 72.9).
These are examples of functions from conjugacy classes which will be called maps in the following. (This should not be confused with the term mapping, cf. Chapter 32.) In GAP, maps are represented by lists. Also each character, each list of element orders, of centralizer orders, or of class lengths are maps, and the list returned by ListPerm (42.5-1), when this function is called with a permutation of classes, is a map.
When maps are constructed without access to a group, often one only knows that the image of a given 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 information, such as centralizer orders, power maps and the restriction of characters, the sets of possible images can be restricted further. In many cases, at the end the images are uniquely determined.
Because of this approach, many functions in this chapter work not only with maps but with parametrized maps (or paramaps for short). More about parametrized maps can be found in Section 73.5.
The implementation follows [Bre91], a description of the main ideas together with several examples can be found in [Bre99].
Several examples in this chapter require the GAP Character Table Library to be available. If it is not yet loaded then we load it now.
gap> LoadPackage( "ctbllib" ); true
The n-th power map of a character table is represented by a list that stores at position i the position of the class containing the n-th powers of the elements in the i-th class. The n-th power map can be composed from the power maps of the prime divisors of n, so usually only power maps for primes are actually stored in the character table.
For an ordinary character table tbl with access to its underlying group G, the p-th power map of tbl can be computed using the identification of the conjugacy classes of G with the classes of tbl. For an ordinary character table without access to a group, in general the p-th power maps (and hence also the element orders) for prime divisors p of the group order are not uniquely determined by the matrix of irreducible characters. So only necessary conditions can be checked in this case, which in general yields only a list of several possibilities for the desired power map. Character tables of the GAP character table library store all p-th power maps for prime divisors p of the group order.
Power maps of Brauer tables can be derived from the power maps of the underlying ordinary tables.
For (computing and) accessing the n-th power map of a character table, PowerMap (73.1-1) can be used; if the n-th power map cannot be uniquely determined then PowerMap (73.1-1) returns fail.
The list of all possible p-th power maps of a table in the sense that certain necessary conditions are satisfied can be computed with PossiblePowerMaps (73.1-2). This provides a default strategy, the subroutines are listed in Section 73.6.
‣ PowerMap( tbl, n[, class] ) | ( operation ) |
‣ PowerMapOp( tbl, n[, class] ) | ( operation ) |
‣ ComputedPowerMaps( tbl ) | ( attribute ) |
Called with first argument a character table tbl and second argument an integer n, PowerMap returns the n-th power map of tbl. This is a list containing at position i the position of the class of n-th powers of the elements in the i-th class of tbl.
If the additional third argument class is present then the position of n-th powers of the class-th class is returned.
If the n-th power map is not uniquely determined by tbl then fail is returned. This can happen only if tbl has no access to its underlying group.
The power maps of tbl that were computed already by PowerMap are stored in tbl as value of the attribute ComputedPowerMaps, the n-th power map at position n. PowerMap checks whether the desired power map is already stored, computes it using the operation PowerMapOp if it is not yet known, and stores it. So methods for the computation of power maps can be installed for the operation PowerMapOp.
gap> tbl:= CharacterTable( "L3(2)" );; gap> ComputedPowerMaps( tbl ); [ , [ 1, 1, 3, 2, 5, 6 ], [ 1, 2, 1, 4, 6, 5 ],,,, [ 1, 2, 3, 4, 1, 1 ] ] gap> PowerMap( tbl, 5 ); [ 1, 2, 3, 4, 6, 5 ] gap> ComputedPowerMaps( tbl ); [ , [ 1, 1, 3, 2, 5, 6 ], [ 1, 2, 1, 4, 6, 5 ],, [ 1, 2, 3, 4, 6, 5 ], , [ 1, 2, 3, 4, 1, 1 ] ] gap> PowerMap( tbl, 137, 2 ); 2
‣ PossiblePowerMaps( tbl, p[, options] ) | ( operation ) |
For the ordinary character table tbl of a group G and a prime integer p, PossiblePowerMaps returns the list of all maps that have the following properties of the p-th power map of tbl. (Representative orders are used only if the OrdersClassRepresentatives (71.9-1) value of tbl is known.
For class i, the centralizer order of the image is a multiple of the i-th centralizer order; if the elements in the i-th class have order coprime to p then the centralizer orders of class i and its image are equal.
Let n be the order of elements in class i. If prime divides n then the images have order n/p; otherwise the images have order n. These criteria are checked in InitPowerMap (73.6-1).
For each character χ of G and each element g in G, the values χ(g^p) and GaloisCyc( χ(g), p ) are algebraic integers that are congruent modulo p; if p does not divide the element order of g then the two values are equal. This congruence is checked for the characters specified below in the discussion of the options argument; For linear characters λ among these characters, the condition χ(g)^p = χ(g^p) is checked. The corresponding function is Congruences (73.6-2).
For each character χ of G, the kernel is a normal subgroup N, and g^p ∈ N for all g ∈ N; moreover, if N has index p in G then g^p ∈ N for all g ∈ G, and if the index of N in G is coprime to p then g^p not ∈ N for each g not ∈ N. These conditions are checked for the kernels of all characters χ specified below, the corresponding function is ConsiderKernels (73.6-3).
If p is larger than the order m of an element g ∈ G then the class of g^p is determined by the power maps for primes dividing the residue of p modulo m. If these power maps are stored in the ComputedPowerMaps (73.1-1) value of tbl then this information is used. This criterion is checked in ConsiderSmallerPowerMaps (73.6-4).
For each character χ of G, the symmetrization ψ defined by ψ(g) = (χ(g)^p - χ(g^p))/p is a character. This condition is checked for the kernels of all characters χ specified below, the corresponding function is PowerMapsAllowedBySymmetrizations (73.6-6).
If tbl is a Brauer table, the possibilities are computed from those for the underlying ordinary table.
The optional argument options, if given, must be a record that may have the following components:
chars:a list of characters which are used for the check of the criteria 3., 4., and 6.; the default is Irr( tbl ),
powermap:a parametrized map which is an approximation of the desired map
decompose:a Boolean; a true value indicates that all constituents of the symmetrizations of chars computed for criterion 6. lie in chars, so the symmetrizations can be decomposed into elements of chars; the default value of decompose is true if chars is not bound and Irr( tbl ) is known, otherwise false,
quick:a Boolean; if true then the subroutines are called with value true for the argument quick; especially, as soon as only one candidate remains this candidate is returned immediately; the default value is false,
parameters:a record with components maxamb, minamb and maxlen which control the subroutine PowerMapsAllowedBySymmetrizations (73.6-6); it only uses characters with current indeterminateness up to maxamb, tests decomposability only for characters with current indeterminateness at least minamb, and admits a branch according to a character only if there is one with at most maxlen possible symmetrizations.
gap> tbl:= CharacterTable( "U4(3).4" );; gap> PossiblePowerMaps( tbl, 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 ] ]
‣ ElementOrdersPowerMap( powermap ) | ( function ) |
Let powermap be a nonempty list containing at position p, if bound, the p-th power map of a character table or group. ElementOrdersPowerMap returns a list of the same length as each entry in powermap, with entry at position i equal to the order of elements in class i if this order is uniquely determined by powermap, and equal to an unknown (see Chapter 74) otherwise.
gap> tbl:= CharacterTable( "U4(3).4" );; gap> known:= ComputedPowerMaps( tbl );; gap> Length( known ); 7 gap> sub:= ShallowCopy( known );; Unbind( sub[7] ); gap> ElementOrdersPowerMap( sub ); [ 1, 2, 3, 3, 3, 4, 4, 5, 6, 6, Unknown(1), Unknown(2), 8, 9, 12, 2, 2, 4, 4, 6, 6, 6, 8, 10, 12, 12, 12, Unknown(3), Unknown(4), 4, 4, 4, 4, 4, 4, 8, 8, 8, 8, 12, 12, 12, 12, 12, 12, 20, 20, 24, 24, Unknown(5), Unknown(6), Unknown(7), Unknown(8) ] gap> ord:= ElementOrdersPowerMap( known ); [ 1, 2, 3, 3, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 12, 2, 2, 4, 4, 6, 6, 6, 8, 10, 12, 12, 12, 14, 14, 4, 4, 4, 4, 4, 4, 8, 8, 8, 8, 12, 12, 12, 12, 12, 12, 20, 20, 24, 24, 28, 28, 28, 28 ] gap> ord = OrdersClassRepresentatives( tbl ); true
‣ PowerMapByComposition( tbl, n ) | ( function ) |
tbl must be a nearly character table, and n a positive integer. If the power maps for all prime divisors of n are stored in the ComputedPowerMaps (73.1-1) list of tbl then PowerMapByComposition returns the n-th power map of tbl. Otherwise fail is returned.
gap> tbl:= CharacterTable( "U4(3).4" );; exp:= Exponent( tbl ); 2520 gap> PowerMapByComposition( tbl, exp ); [ 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 ] gap> Length( ComputedPowerMaps( tbl ) ); 7 gap> PowerMapByComposition( tbl, 11 ); fail gap> PowerMap( tbl, 11 );; gap> PowerMapByComposition( tbl, 11 ); [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 25, 27, 28, 29, 31, 30, 33, 32, 35, 34, 37, 36, 39, 38, 41, 40, 43, 42, 45, 44, 47, 46, 49, 48, 51, 50, 53, 52 ]
The permutation group of matrix automorphisms (see MatrixAutomorphisms (71.22-1)) acts on the possible power maps returned by PossiblePowerMaps (73.1-2) by permuting a list via Permuted (21.20-17) and then mapping the images via OnPoints (41.2-1). Note that by definition, the group of table automorphisms acts trivially.
‣ OrbitPowerMaps( map, permgrp ) | ( function ) |
returns the orbit of the power map map under the action of the permutation group permgrp via a combination of Permuted (21.20-17) and OnPoints (41.2-1).
‣ RepresentativesPowerMaps( listofmaps, permgrp ) | ( function ) |
returns a list of orbit representatives of the power maps in the list listofmaps under the action of the permutation group permgrp via a combination of Permuted (21.20-17) and OnPoints (41.2-1).
gap> tbl:= CharacterTable( "3.McL" );; gap> grp:= MatrixAutomorphisms( Irr( tbl ) ); Size( grp ); <permutation group with 5 generators> 32 gap> poss:= PossiblePowerMaps( CharacterTable( "3.McL" ), 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> reps:= RepresentativesPowerMaps( poss, grp ); [ [ 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> orb:= OrbitPowerMaps( reps[1], grp ); [ [ 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 ] ] gap> Parametrized( orb ); [ 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 ], [ 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 ]
For a group G and a subgroup H of G, the fusion map between the character table of H and the character table of G is represented by a list that stores at position i the position of the i-th class of the table of H in the classes list of the table of G.
For ordinary character tables tbl1 and tbl2 of H and G, with access to the groups H and G, the class fusion between tbl1 and tbl2 can be computed using the identifications of the conjugacy classes of H with the classes of tbl1 and the conjugacy classes of G with the classes of tbl2. For two ordinary character tables without access to an underlying group, or in the situation that the group stored in tbl1 is not physically a subgroup of the group stored in tbl2 but an isomorphic copy, in general the class fusion is not uniquely determined by the information stored on the tables such as irreducible characters and power maps. So only necessary conditions can be checked in this case, which in general yields only a list of several possibilities for the desired class fusion. Character tables of the GAP character table library store various class fusions that are regarded as important, for example fusions from maximal subgroups (see ComputedClassFusions (73.3-2) and Maxes (CTblLib: Maxes) in the manual for the GAP Character Table Library).
Class fusions between Brauer tables can be derived from the class fusions between the underlying ordinary tables. The class fusion from a Brauer table to the underlying ordinary table is stored when the Brauer table is constructed from the ordinary table, so no method is needed to compute such a fusion.
For (computing and) accessing the class fusion between two character tables, FusionConjugacyClasses (73.3-1) can be used; if the class fusion cannot be uniquely determined then FusionConjugacyClasses (73.3-1) returns fail.
The list of all possible class fusion between two tables in the sense that certain necessary conditions are satisfied can be computed with PossibleClassFusions (73.3-6). This provides a default strategy, the subroutines are listed in Section 73.7.
It should be noted that all the following functions except FusionConjugacyClasses (73.3-1) deal only with the situation of class fusions from subgroups. The computation of factor fusions from a character table to the table of a factor group is not dealt with here. Since the ordinary character table of a group G determines the character tables of all factor groups of G, the factor fusion to a given character table of a factor group of G is determined up to table automorphisms (see AutomorphismsOfTable (71.9-4)) once the class positions of the kernel of the natural epimorphism have been fixed.
‣ FusionConjugacyClasses( tbl1, tbl2 ) | ( operation ) |
‣ FusionConjugacyClasses( H, G ) | ( operation ) |
‣ FusionConjugacyClasses( hom[, tbl1, tbl2] ) | ( operation ) |
‣ FusionConjugacyClassesOp( tbl1, tbl2 ) | ( operation ) |
‣ FusionConjugacyClassesOp( hom ) | ( attribute ) |
Called with two character tables tbl1 and tbl2, FusionConjugacyClasses returns the fusion of conjugacy classes between tbl1 and tbl2. (If one of the tables is a Brauer table, it will delegate this task to the underlying ordinary table.)
Called with two groups H and G where H is a subgroup of G, FusionConjugacyClasses returns the fusion of conjugacy classes between H and G. This is done by delegating to the ordinary character tables of H and G, since class fusions are stored only for character tables and not for groups.
Note that the returned class fusion refers to the ordering of conjugacy classes in the character tables if the arguments are character tables and to the ordering of conjugacy classes in the groups if the arguments are groups (see ConjugacyClasses (71.6-2)).
Called with a group homomorphism hom, FusionConjugacyClasses returns the fusion of conjugacy classes between the preimage and the image of hom; contrary to the two cases above, also factor fusions can be handled by this variant. If hom is the only argument then the class fusion refers to the ordering of conjugacy classes in the groups. If the character tables of preimage and image are given as tbl1 and tbl2, respectively (each table with its group stored), then the fusion refers to the ordering of classes in these tables.
If no class fusion exists or if the class fusion is not uniquely determined, fail is returned; this may happen when FusionConjugacyClasses is called with two character tables that do not know compatible underlying groups.
Methods for the computation of class fusions can be installed for the operation FusionConjugacyClassesOp.
gap> s4:= SymmetricGroup( 4 ); Sym( [ 1 .. 4 ] ) gap> tbls4:= CharacterTable( s4 );; gap> d8:= SylowSubgroup( s4, 2 ); Group([ (1,2), (3,4), (1,3)(2,4) ]) gap> FusionConjugacyClasses( d8, s4 ); [ 1, 2, 3, 3, 5 ] gap> tbls5:= CharacterTable( "S5" );; gap> FusionConjugacyClasses( CharacterTable( "A5" ), tbls5 ); [ 1, 2, 3, 4, 4 ] gap> FusionConjugacyClasses(CharacterTable("A5"), CharacterTable("J1")); fail gap> PossibleClassFusions(CharacterTable("A5"), CharacterTable("J1")); [ [ 1, 2, 3, 4, 5 ], [ 1, 2, 3, 5, 4 ] ]
‣ ComputedClassFusions( tbl ) | ( attribute ) |
The class fusions from the character table tbl that have been computed already by FusionConjugacyClasses (73.3-1) or explicitly stored by StoreFusion (73.3-4) are stored in the ComputedClassFusions list of tbl1. Each entry of this list is a record with the following components.
namethe Identifier (71.9-8) value of the character table to which the fusion maps,
mapthe list of positions of image classes,
text (optional)a string giving additional information about the fusion map, for example whether the map is uniquely determined by the character tables,
specification (optional, rarely used)a value that distinguishes different fusions between the same tables.
Note that stored fusion maps may differ from the maps returned by GetFusionMap (73.3-3) and the maps entered by StoreFusion (73.3-4) if the table destination has a nonidentity ClassPermutation (71.21-5) value. So if one fetches a fusion map from a table tbl1 to a table tbl2 via access to the data in the ComputedClassFusions list of tbl1 then the stored value must be composed with the ClassPermutation (71.21-5) value of tbl2 in order to obtain the correct class fusion. (If one handles fusions only via GetFusionMap (73.3-3) and StoreFusion (73.3-4) then this adjustment is made automatically.)
Fusions are identified via the Identifier (71.9-8) value of the destination table and not by this table itself because many fusions between character tables in the GAP character table library are stored on library tables, and it is not desirable to load together with a library table also all those character tables that occur as destinations of fusions from this table.
For storing fusions and accessing stored fusions, see also GetFusionMap (73.3-3), StoreFusion (73.3-4). For accessing the identifiers of tables that store a fusion into a given character table, see NamesOfFusionSources (73.3-5).
‣ GetFusionMap( source, destination[, specification] ) | ( function ) |
For two ordinary character tables source and destination, GetFusionMap checks whether the ComputedClassFusions (73.3-2) list of source contains a record with name component Identifier( destination ), and returns the map component of the first such record. GetFusionMap( source, destination, specification ) fetches that fusion map for which the record additionally has the specification component specification.
If both source and destination are Brauer tables, first the same is done, and if no fusion map was found then GetFusionMap looks whether a fusion map between the ordinary tables is stored; if so then the fusion map between source and destination is stored on source, and then returned.
If no appropriate fusion is found, GetFusionMap returns fail. For the computation of class fusions, see FusionConjugacyClasses (73.3-1).
‣ StoreFusion( source, fusion, destination ) | ( function ) |
For two character tables source and destination, StoreFusion stores the fusion fusion from source to destination in the ComputedClassFusions (73.3-2) list of source, and adds the Identifier (71.9-8) string of destination to the NamesOfFusionSources (73.3-5) list of destination.
fusion can either be a fusion map (that is, the list of positions of the image classes) or a record as described in ComputedClassFusions (73.3-2).
If fusions to destination are already stored on source then another fusion can be stored only if it has a record component specification that distinguishes it from the stored fusions. In the case of such an ambiguity, StoreFusion raises an error.
gap> tbld8:= CharacterTable( d8 );; gap> ComputedClassFusions( tbld8 ); [ rec( map := [ 1, 2, 3, 3, 5 ], name := "CT1" ) ] gap> Identifier( tbls4 ); "CT1" gap> GetFusionMap( tbld8, tbls4 ); [ 1, 2, 3, 3, 5 ] gap> GetFusionMap( tbls4, tbls5 ); fail gap> poss:= PossibleClassFusions( tbls4, tbls5 ); [ [ 1, 5, 2, 3, 6 ] ] gap> StoreFusion( tbls4, poss[1], tbls5 ); gap> GetFusionMap( tbls4, tbls5 ); [ 1, 5, 2, 3, 6 ]
‣ NamesOfFusionSources( tbl ) | ( attribute ) |
For a character table tbl, NamesOfFusionSources returns the list of identifiers of all those character tables that are known to have fusions to tbl stored. The NamesOfFusionSources value is updated whenever a fusion to tbl is stored using StoreFusion (73.3-4).
gap> NamesOfFusionSources( tbls4 ); [ "CT2" ] gap> Identifier( CharacterTable( d8 ) ); "CT2"
‣ PossibleClassFusions( subtbl, tbl[, options] ) | ( operation ) |
For two ordinary character tables subtbl and tbl of the groups H and G, PossibleClassFusions returns the list of all maps that have the following properties of class fusions from subtbl to tbl.
For class i, the centralizer order of the image in G is a multiple of the i-th centralizer order in H, and the element orders in the i-th class and its image are equal. These criteria are checked in InitFusion (73.7-1).
The class fusion commutes with power maps. This is checked using TestConsistencyMaps (73.5-12).
If the permutation character of G corresponding to the action of G on the cosets of H is specified (see the discussion of the options argument below) then it prescribes for each class C of G the number of elements of H fusing into C. The corresponding function is CheckPermChar (73.7-2).
The table automorphisms of tbl (see AutomorphismsOfTable (71.9-4)) are used in order to compute only orbit representatives. (But note that the list returned by PossibleClassFusions contains the full orbits.)
For each character χ of G, the restriction to H via the class fusion is a character of H. This condition is checked for all characters specified below, the corresponding function is FusionsAllowedByRestrictions (73.7-4).
The class multiplication coefficients in subtbl do not exceed the corresponding coefficients in tbl. This is checked in ConsiderStructureConstants (73.3-7), see also the comment on the parameter verify below.
If subtbl and tbl are Brauer tables then the possibilities are computed from those for the underlying ordinary tables.
The optional argument options must be a record that may have the following components:
charsa list of characters of tbl which are used for the check of 5.; the default is Irr( tbl ),
subcharsa list of characters of subtbl which are constituents of the restrictions of chars, the default is Irr( subtbl ),
fusionmapa parametrized map which is an approximation of the desired map,
decomposea Boolean; a true value indicates that all constituents of the restrictions of chars computed for criterion 5. lie in subchars, so the restrictions can be decomposed into elements of subchars; the default value of decompose is true if subchars is not bound and Irr( subtbl ) is known, otherwise false,
permchar(a values list of) a permutation character; only those fusions affording that permutation character are computed,
quicka Boolean; if true then the subroutines are called with value true for the argument quick; especially, as soon as only one possibility remains then this possibility is returned immediately; the default value is false (note that in situations where the group of tbl has no subgroups with character table subtbl, it may happen that setting quick to true causes PossibleClassFusions to return solutions, whereas the value false yields an empty list),
verifya Boolean; if false then ConsiderStructureConstants (73.3-7) is called only if more than one orbit of possible class fusions exists, under the action of the groups of table automorphisms; the default value is false (because the computation of the structure constants is usually very time consuming, compared with checking the other criteria),
parametersa record with components maxamb, minamb and maxlen (and perhaps some optional components) which control the subroutine FusionsAllowedByRestrictions (73.7-4); it only uses characters with current indeterminateness up to maxamb, tests decomposability only for characters with current indeterminateness at least minamb, and admits a branch according to a character only if there is one with at most maxlen possible restrictions.
gap> subtbl:= CharacterTable( "U3(3)" );; tbl:= CharacterTable( "J4" );; gap> PossibleClassFusions( subtbl, tbl ); [ [ 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 ] ]
‣ ConsiderStructureConstants( subtbl, tbl, fusions, quick ) | ( function ) |
Let subtbl and tbl be ordinary character tables and fusions be a list of possible class fusions from subtbl to tbl. ConsiderStructureConstants returns the list of those maps σ in fusions with the property that for all triples (i,j,k) of class positions, ClassMultiplicationCoefficient( subtbl, i, j, k ) is not bigger than ClassMultiplicationCoefficient( tbl, σ[i], σ[j], σ[k] ); see ClassMultiplicationCoefficient (71.12-7) for the definition of class multiplication coefficients/structure constants.
The argument quick must be a Boolean; if it is true then only those triples are checked for which at least two entries in fusions have different images.
The permutation groups of table automorphisms (see AutomorphismsOfTable (71.9-4)) of the subgroup table subtbl and the supergroup table tbl act on the possible class fusions from subtbl to tbl that are returned by PossibleClassFusions (73.3-6), the former by permuting a list via Permuted (21.20-17), the latter by mapping the images via OnPoints (41.2-1).
If a set of possible fusions with certain properties was computed that are not invariant under the full groups of table automorphisms then only a smaller group acts on this set. This may happen for example if a permutation character or if an explicit approximation of the fusion map was prescribed in the call of PossibleClassFusions (73.3-6).
‣ OrbitFusions( subtblautomorphisms, fusionmap, tblautomorphisms ) | ( function ) |
returns the orbit of the class fusion map fusionmap under the actions of the permutation groups subtblautomorphisms and tblautomorphisms of automorphisms of the character table of the subgroup and the supergroup, respectively.
‣ RepresentativesFusions( subtbl, listofmaps, tbl ) | ( function ) |
Let listofmaps be a list of class fusions from the character table subtbl to the character table tbl. RepresentativesFusions returns a list of orbit representatives of the class fusions under the action of maximal admissible subgroups of the table automorphism groups of these character tables.
Instead of the character tables subtbl and tbl, also the permutation groups of their table automorphisms (see AutomorphismsOfTable (71.9-4)) may be entered.
gap> fus:= GetFusionMap( subtbl, tbl ); [ 1, 2, 4, 4, 5, 5, 6, 10, 12, 13, 14, 14, 21, 21 ] gap> orb:= OrbitFusions( AutomorphismsOfTable( subtbl ), fus, > AutomorphismsOfTable( tbl ) ); [ [ 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 ] ] gap> rep:= RepresentativesFusions( subtbl, orb, tbl ); [ [ 1, 2, 4, 4, 5, 5, 6, 10, 12, 13, 14, 14, 21, 21 ] ]
A parametrized map is a list whose i-th entry is either unbound (which means that nothing is known about the image(s) of the i-th class) or the image of the i-th class (i.e., an integer for fusion maps, power maps, element orders etc., and a cyclotomic for characters), or a list of possible images of the i-th class. In this sense, maps are special parametrized maps. We often identify a parametrized map paramap with the set of all maps map with the property that either map[i] = paramap[i] or map[i] is contained in the list paramap[i]; we say then that map is contained in paramap.
This definition implies that parametrized maps cannot be used to describe sets of maps where lists are possible images. An exception are strings which naturally arise as images when class names are considered. So strings and lists of strings are allowed in parametrized maps, and character constants (see Chapter 27) are not allowed in maps.
‣ CompositionMaps( paramap2, paramap1[, class] ) | ( function ) |
The composition of two parametrized maps paramap1, paramap2 is defined as the parametrized map comp that contains all compositions f_2 ∘ f_1 of elements f_1 of paramap1 and f_2 of paramap2. For example, the composition of a character χ of a group G by a parametrized class fusion map from a subgroup H to G is the parametrized map that contains all restrictions of χ by elements of the parametrized fusion map.
CompositionMaps(paramap2, paramap1) is a parametrized map with entry CompositionMaps(paramap2, paramap1, class) at position class. If paramap1[class] is an integer then CompositionMaps(paramap2, paramap1, class) is equal to paramap2[ paramap1[ class ] ]. Otherwise it is the union of paramap2[i] for i in paramap1[ class ].
gap> map1:= [ 1, [ 2 .. 4 ], [ 4, 5 ], 1 ];; gap> map2:= [ [ 1, 2 ], 2, 2, 3, 3 ];; gap> CompositionMaps( map2, map1 ); [ [ 1, 2 ], [ 2, 3 ], 3, [ 1, 2 ] ] gap> CompositionMaps( map1, map2 ); [ [ 1, 2, 3, 4 ], [ 2 .. 4 ], [ 2 .. 4 ], [ 4, 5 ], [ 4, 5 ] ]
‣ InverseMap( paramap ) | ( function ) |
For a parametrized map paramap, InverseMap returns a mutable parametrized map whose i-th entry is unbound if i is not in the image of paramap, equal to j if i is (in) the image of paramap[j] exactly for j, and equal to the set of all preimages of i under paramap otherwise.
We have CompositionMaps( paramap, InverseMap( paramap ) ) the identity map.
gap> tbl:= CharacterTable( "2.A5" );; f:= CharacterTable( "A5" );; gap> fus:= GetFusionMap( tbl, f ); [ 1, 1, 2, 3, 3, 4, 4, 5, 5 ] gap> inv:= InverseMap( fus ); [ [ 1, 2 ], 3, [ 4, 5 ], [ 6, 7 ], [ 8, 9 ] ] gap> CompositionMaps( fus, inv ); [ 1, 2, 3, 4, 5 ] gap> # transfer a power map ``up'' to the factor group gap> pow:= PowerMap( tbl, 2 ); [ 1, 1, 2, 4, 4, 8, 8, 6, 6 ] gap> CompositionMaps( fus, CompositionMaps( pow, inv ) ); [ 1, 1, 3, 5, 4 ] gap> last = PowerMap( f, 2 ); true gap> # transfer a power map of the factor group ``down'' to the group gap> CompositionMaps( inv, CompositionMaps( PowerMap( f, 2 ), fus ) ); [ [ 1, 2 ], [ 1, 2 ], [ 1, 2 ], [ 4, 5 ], [ 4, 5 ], [ 8, 9 ], [ 8, 9 ], [ 6, 7 ], [ 6, 7 ] ]
‣ ProjectionMap( fusionmap ) | ( function ) |
For a map fusionmap, ProjectionMap returns a parametrized map whose i-th entry is unbound if i is not in the image of fusionmap, and equal to j if j is the smallest position such that i is the image of fusionmap[j].
We have CompositionMaps( fusionmap, ProjectionMap( fusionmap ) ) the identity map, i.e., first projecting and then fusing yields the identity. Note that fusionmap must not be a parametrized map.
gap> ProjectionMap( [ 1, 1, 1, 2, 2, 2, 3, 4, 5, 5, 5, 6, 6, 6 ] ); [ 1, 4, 7, 8, 9, 12 ]
‣ Indirected( character, paramap ) | ( function ) |
For a map character and a parametrized map paramap, Indirected returns a parametrized map whose entry at position i is character[ paramap[i] ] if paramap[i] is an integer, and an unknown (see Chapter 74) otherwise.
gap> tbl:= CharacterTable( "M12" );; gap> fus:= [ 1, 3, 4, [ 6, 7 ], 8, 10, [ 11, 12 ], [ 11, 12 ], > [ 14, 15 ], [ 14, 15 ] ];; gap> List( Irr( tbl ){ [ 1 .. 6 ] }, x -> Indirected( x, fus ) ); [ [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ], [ 11, 3, 2, Unknown(9), 1, 0, Unknown(10), Unknown(11), 0, 0 ], [ 11, 3, 2, Unknown(12), 1, 0, Unknown(13), Unknown(14), 0, 0 ], [ 16, 0, -2, 0, 1, 0, 0, 0, Unknown(15), Unknown(16) ], [ 16, 0, -2, 0, 1, 0, 0, 0, Unknown(17), Unknown(18) ], [ 45, -3, 0, 1, 0, 0, -1, -1, 1, 1 ] ]
‣ Parametrized( list ) | ( function ) |
For a list list of (parametrized) maps of the same length, Parametrized returns the smallest parametrized map containing all elements of list.
Parametrized is the inverse function to ContainedMaps (73.5-6).
gap> Parametrized( [ [ 1, 2, 3, 4, 5 ], [ 1, 3, 2, 4, 5 ], > [ 1, 2, 3, 4, 6 ] ] ); [ 1, [ 2, 3 ], [ 2, 3 ], 4, [ 5, 6 ] ]
‣ ContainedMaps( paramap ) | ( function ) |
For a parametrized map paramap, ContainedMaps returns the set of all maps contained in paramap.
ContainedMaps is the inverse function to Parametrized (73.5-5) in the sense that Parametrized( ContainedMaps( paramap ) ) is equal to paramap.
gap> ContainedMaps( [ 1, [ 2, 3 ], [ 2, 3 ], 4, [ 5, 6 ] ] ); [ [ 1, 2, 2, 4, 5 ], [ 1, 2, 2, 4, 6 ], [ 1, 2, 3, 4, 5 ], [ 1, 2, 3, 4, 6 ], [ 1, 3, 2, 4, 5 ], [ 1, 3, 2, 4, 6 ], [ 1, 3, 3, 4, 5 ], [ 1, 3, 3, 4, 6 ] ]
‣ UpdateMap( character, paramap, indirected ) | ( function ) |
Let character be a map, paramap a parametrized map, and indirected a parametrized map that is contained in CompositionMaps( character, paramap ).
Then UpdateMap changes paramap to the parametrized map containing exactly the maps whose composition with character is equal to indirected.
If a contradiction is detected then false is returned immediately, otherwise true.
gap> subtbl:= CharacterTable("S4(4).2");; tbl:= CharacterTable("He");; gap> fus:= InitFusion( subtbl, tbl );; gap> fus; [ 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> chi:= Irr( tbl )[2]; Character( CharacterTable( "He" ), [ 51, 11, 3, 6, 0, 3, 3, -1, 1, 2, 0, 3*E(7)+3*E(7)^2+3*E(7)^4, 3*E(7)^3+3*E(7)^5+3*E(7)^6, 2, E(7)+E(7)^2+2*E(7)^3+E(7)^4+2*E(7)^5+2*E(7)^6, 2*E(7)+2*E(7)^2+E(7)^3+2*E(7)^4+E(7)^5+E(7)^6, 1, 1, 0, 0, -E(7)-E(7)^2-E(7)^4, -E(7)^3-E(7)^5-E(7)^6, E(7)+E(7)^2+E(7)^4, E(7)^3+E(7)^5+E(7)^6, 1, 0, 0, -1, -1, 0, 0, E(7)+E(7)^2+E(7)^4, E(7)^3+E(7)^5+E(7)^6 ] ) gap> filt:= Filtered( Irr( subtbl ), x -> x[1] = 50 ); [ Character( CharacterTable( "S4(4).2" ), [ 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 ] ), Character( CharacterTable( "S4(4).2" ), [ 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( chi, fus, filt[1] + TrivialCharacter( subtbl ) ); true 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 ] ]
‣ MeetMaps( paramap1, paramap2 ) | ( function ) |
For two parametrized maps paramap1 and paramap2, MeetMaps changes paramap1 such that the image of class i is the intersection of paramap1[i] and paramap2[i].
If this implies that no images remain for a class, the position of such a class is returned. If no such inconsistency occurs, MeetMaps returns true.
gap> map1:= [ [ 1, 2 ], [ 3, 4 ], 5, 6, [ 7, 8, 9 ] ];; gap> map2:= [ [ 1, 3 ], [ 3, 4 ], [ 5, 6 ], 6, [ 8, 9, 10 ] ];; gap> MeetMaps( map1, map2 ); map1; true [ 1, [ 3, 4 ], 5, 6, [ 8, 9 ] ]
‣ CommutativeDiagram( paramap1, paramap2, paramap3, paramap4[, improvements] ) | ( function ) |
Let paramap1, paramap2, paramap3, paramap4 be parametrized maps covering parametrized maps f_1, f_2, f_3, f_4 with the property that CompositionMaps( f_2, f_1 ) is equal to CompositionMaps( f_4, f_3 ).
CommutativeDiagram checks this consistency, and changes the arguments such that all possible images are removed that cannot occur in the parametrized maps f_i.
The return value is fail if an inconsistency was found. Otherwise a record with the components imp1, imp2, imp3, imp4 is returned, each bound to the list of positions where the corresponding parametrized map was changed,
The optional argument improvements must be a record with components imp1, imp2, imp3, imp4. If such a record is specified then only diagrams are considered where entries of the i-th component occur as preimages of the i-th parametrized map.
When an inconsistency is detected, CommutativeDiagram immediately returns fail. Otherwise a record is returned that contains four lists imp1, ..., imp4: The i-th component is the list of classes where the i-th argument was changed.
gap> map1:= [[ 1, 2, 3 ], [ 1, 3 ]];; map2:= [[ 1, 2 ], 1, [ 1, 3 ]];; gap> map3:= [ [ 2, 3 ], 3 ];; map4:= [ , 1, 2, [ 1, 2 ] ];; gap> imp:= CommutativeDiagram( map1, map2, map3, map4 ); rec( imp1 := [ 2 ], imp2 := [ 1 ], imp3 := [ ], imp4 := [ ] ) gap> map1; map2; map3; map4; [ [ 1, 2, 3 ], 1 ] [ 2, 1, [ 1, 3 ] ] [ [ 2, 3 ], 3 ] [ , 1, 2, [ 1, 2 ] ] gap> imp2:= CommutativeDiagram( map1, map2, map3, map4, imp ); rec( imp1 := [ ], imp2 := [ ], imp3 := [ ], imp4 := [ ] )
‣ CheckFixedPoints( inside1, between, inside2 ) | ( function ) |
Let inside1, between, inside2 be parametrized maps, where between is assumed to map each fixed point of inside1 (that is, inside1[i] = i) to a fixed point of inside2 (that is, between[i] is either an integer that is fixed by inside2 or a list that has nonempty intersection with the union of its images under inside2). CheckFixedPoints changes between and inside2 by removing all those entries violate this condition.
When an inconsistency is detected, CheckFixedPoints immediately returns fail. Otherwise the list of positions is returned where changes occurred.
gap> subtbl:= CharacterTable( "L4(3).2_2" );; gap> tbl:= CharacterTable( "O7(3)" );; gap> fus:= InitFusion( subtbl, tbl );; fus{ [ 48, 49 ] }; [ [ 54, 55, 56, 57 ], [ 54, 55, 56, 57 ] ] gap> CheckFixedPoints( ComputedPowerMaps( subtbl )[5], fus, > ComputedPowerMaps( tbl )[5] ); [ 48, 49 ] gap> fus{ [ 48, 49 ] }; [ [ 56, 57 ], [ 56, 57 ] ]
‣ TransferDiagram( inside1, between, inside2[, improvements] ) | ( function ) |
Let inside1, between, inside2 be parametrized maps covering parametrized maps m_1, f, m_2 with the property that CompositionMaps( m_2, f ) is equal to CompositionMaps( f, m_1 ).
TransferDiagram checks this consistency, and changes the arguments such that all possible images are removed that cannot occur in the parametrized maps m_i and f.
So TransferDiagram is similar to CommutativeDiagram (73.5-9), but between occurs twice in each diagram checked.
If a record improvements with fields impinside1, impbetween, and impinside2 is specified, only those diagrams with elements of impinside1 as preimages of inside1, elements of impbetween as preimages of between or elements of impinside2 as preimages of inside2 are considered.
When an inconsistency is detected, TransferDiagram immediately returns fail. Otherwise a record is returned that contains three lists impinside1, impbetween, and impinside2 of positions where the arguments were changed.
gap> subtbl:= CharacterTable( "2F4(2)" );; tbl:= CharacterTable( "Ru" );; gap> fus:= InitFusion( subtbl, tbl );; gap> permchar:= Sum( Irr( tbl ){ [ 1, 5, 6 ] } );; gap> CheckPermChar( subtbl, tbl, 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> tr:= TransferDiagram(PowerMap( subtbl, 2), fus, PowerMap(tbl, 2)); rec( impbetween := [ 12, 23 ], impinside1 := [ ], impinside2 := [ ] ) gap> tr:= TransferDiagram(PowerMap(subtbl, 3), fus, PowerMap( tbl, 3 )); rec( impbetween := [ 14, 24, 25 ], impinside1 := [ ], impinside2 := [ ] ) gap> tr:= TransferDiagram( PowerMap(subtbl, 3), fus, PowerMap(tbl, 3), > tr ); rec( impbetween := [ ], impinside1 := [ ], 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 ]
‣ TestConsistencyMaps( powermap1, fusionmap, powermap2[, fusimp] ) | ( function ) |
Let powermap1 and powermap2 be lists of parametrized maps, and fusionmap a parametrized map, such that for each i, the i-th entry in powermap1, fusionmap, and the i-th entry in powermap2 (if bound) are valid arguments for TransferDiagram (73.5-11). So a typical situation for applying TestConsistencyMaps is that fusionmap is an approximation of a class fusion, and powermap1, powermap2 are the lists of power maps of the subgroup and the group.
TestConsistencyMaps repeatedly applies TransferDiagram (73.5-11) to these arguments for all i until no more changes occur.
If a list fusimp is specified then only those diagrams with elements of fusimp as preimages of fusionmap are considered.
When an inconsistency is detected, TestConsistencyMaps immediately returns false. Otherwise true is returned.
gap> subtbl:= CharacterTable( "2F4(2)" );; tbl:= CharacterTable( "Ru" );; gap> fus:= InitFusion( subtbl, tbl );; gap> permchar:= Sum( Irr( tbl ){ [ 1, 5, 6 ] } );; gap> CheckPermChar( subtbl, tbl, 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( ComputedPowerMaps( subtbl ), fus, > ComputedPowerMaps( tbl ) ); 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
‣ Indeterminateness( paramap ) | ( function ) |
For a parametrized map paramap, Indeterminateness returns the number of maps contained in paramap, that is, the product of lengths of lists in paramap denoting lists of several images.
gap> Indeterminateness([ 1, [ 2, 3 ], [ 4, 5 ], [ 6, 7, 8, 9, 10 ], 11 ]); 20
‣ PrintAmbiguity( list, paramap ) | ( function ) |
For each map in the list list, PrintAmbiguity prints its position in list, the indeterminateness (see Indeterminateness (73.5-13)) of the composition with the parametrized map paramap, and the list of positions where a list of images occurs in this composition.
gap> paramap:= [ 1, [ 2, 3 ], [ 3, 4 ], [ 2, 3, 4 ], 5 ];; gap> list:= [ [ 1, 1, 1, 1, 1 ], [ 1, 1, 2, 2, 3 ], [ 1, 2, 3, 4, 5 ] ];; gap> PrintAmbiguity( list, paramap ); 1 1 [ ] 2 4 [ 2, 4 ] 3 12 [ 2, 3, 4 ]
‣ ContainedSpecialVectors( tbl, chars, paracharacter, func ) | ( function ) |
‣ IntScalarProducts( tbl, chars, candidate ) | ( function ) |
‣ NonnegIntScalarProducts( tbl, chars, candidate ) | ( function ) |
‣ ContainedPossibleVirtualCharacters( tbl, chars, paracharacter ) | ( function ) |
‣ ContainedPossibleCharacters( tbl, chars, paracharacter ) | ( function ) |
Let tbl be an ordinary character table, chars a list of class functions (or values lists), paracharacter a parametrized class function of tbl, and func a function that expects the three arguments tbl, chars, and a values list of a class function, and that returns either true or false.
ContainedSpecialVectors returns the list of all those elements vec of paracharacter that have integral norm, have integral scalar product with the principal character of tbl, and that satisfy func( tbl, chars, vec ) = true.
Two special cases of func are the check whether the scalar products in tbl between the vector vec and all lists in chars are integers or nonnegative integers, respectively. These functions are accessible as global variables IntScalarProducts and NonnegIntScalarProducts, and ContainedPossibleVirtualCharacters and ContainedPossibleCharacters provide access to these special cases of ContainedSpecialVectors.
gap> subtbl:= CharacterTable( "HSM12" );; tbl:= CharacterTable( "HS" );; gap> fus:= InitFusion( subtbl, tbl );; gap> rest:= CompositionMaps( Irr( tbl )[8], fus ); [ 231, [ -9, 7 ], [ -9, 7 ], [ -9, 7 ], 6, 15, 15, [ -1, 15 ], [ -1, 15 ], 1, [ 1, 6 ], [ 1, 6 ], [ 1, 6 ], [ 1, 6 ], [ -2, 0 ], [ 1, 2 ], [ 1, 2 ], [ 1, 2 ], 0, 0, 1, 0, 0, 0, 0 ] gap> irr:= Irr( subtbl );; gap> # no further condition gap> cont1:= ContainedSpecialVectors( subtbl, irr, rest, > function( tbl, chars, vec ) return true; end );; gap> Length( cont1 ); 24 gap> # require scalar products to be integral gap> cont2:= ContainedSpecialVectors( subtbl, irr, rest, > IntScalarProducts ); [ [ 231, 7, -9, -9, 6, 15, 15, -1, -1, 1, 6, 6, 1, 1, -2, 1, 2, 2, 0, 0, 1, 0, 0, 0, 0 ], [ 231, 7, -9, 7, 6, 15, 15, -1, -1, 1, 6, 6, 1, 1, -2, 1, 2, 2, 0, 0, 1, 0, 0, 0, 0 ], [ 231, 7, -9, -9, 6, 15, 15, 15, 15, 1, 6, 6, 1, 1, -2, 1, 2, 2, 0, 0, 1, 0, 0, 0, 0 ], [ 231, 7, -9, 7, 6, 15, 15, 15, 15, 1, 6, 6, 1, 1, -2, 1, 2, 2, 0, 0, 1, 0, 0, 0, 0 ] ] gap> # additionally require scalar products to be nonnegative gap> cont3:= ContainedSpecialVectors( subtbl, irr, rest, > NonnegIntScalarProducts ); [ [ 231, 7, -9, -9, 6, 15, 15, -1, -1, 1, 6, 6, 1, 1, -2, 1, 2, 2, 0, 0, 1, 0, 0, 0, 0 ], [ 231, 7, -9, 7, 6, 15, 15, -1, -1, 1, 6, 6, 1, 1, -2, 1, 2, 2, 0, 0, 1, 0, 0, 0, 0 ] ] gap> cont2 = ContainedPossibleVirtualCharacters( subtbl, irr, rest ); true gap> cont3 = ContainedPossibleCharacters( subtbl, irr, rest ); true
‣ CollapsedMat( mat, maps ) | ( function ) |
is a record with the components
fusionfusion that collapses those columns of mat that are equal in mat and also for all maps in the list maps,
matthe image of mat under that fusion.
gap> mat:= [ [ 1, 1, 1, 1 ], [ 2, -1, 0, 0 ], [ 4, 4, 1, 1 ] ];; gap> coll:= CollapsedMat( mat, [] ); rec( fusion := [ 1, 2, 3, 3 ], mat := [ [ 1, 1, 1 ], [ 2, -1, 0 ], [ 4, 4, 1 ] ] ) gap> List( last.mat, x -> x{ last.fusion } ) = mat; true gap> coll:= CollapsedMat( mat, [ [ 1, 1, 1, 2 ] ] ); rec( fusion := [ 1, 2, 3, 4 ], mat := [ [ 1, 1, 1, 1 ], [ 2, -1, 0, 0 ], [ 4, 4, 1, 1 ] ] )
‣ ContainedDecomposables( constituents, moduls, parachar, func ) | ( function ) |
‣ ContainedCharacters( tbl, constituents, parachar ) | ( function ) |
For these functions, let constituents be a list of rational class functions, moduls a list of positive integers, parachar a parametrized rational class function, func a function that returns either true or false when called with (a values list of) a class function, and tbl a character table.
ContainedDecomposables returns the set of all elements χ of parachar that satisfy func( χ ) = true and that lie in the ℤ-lattice spanned by constituents, modulo moduls. The latter means they lie in the ℤ-lattice spanned by constituents and the set { moduls[i] ⋅ e_i; 1 ≤ i ≤ n } where n is the length of parachar and e_i is the i-th standard basis vector.
One application of ContainedDecomposables is the following. constituents is a list of (values lists of) rational characters of an ordinary character table tbl, moduls is the list of centralizer orders of tbl (see SizesCentralizers (71.9-2)), and func checks whether a vector in the lattice mentioned above has nonnegative integral scalar product in tbl with all entries of constituents. This situation is handled by ContainedCharacters. Note that the entries of the result list are not necessary linear combinations of constituents, and they are not necessarily characters of tbl.
gap> subtbl:= CharacterTable( "HSM12" );; tbl:= CharacterTable( "HS" );; gap> rat:= RationalizedMat( Irr( subtbl ) );; gap> fus:= InitFusion( subtbl, tbl );; gap> rest:= CompositionMaps( Irr( tbl )[8], fus ); [ 231, [ -9, 7 ], [ -9, 7 ], [ -9, 7 ], 6, 15, 15, [ -1, 15 ], [ -1, 15 ], 1, [ 1, 6 ], [ 1, 6 ], [ 1, 6 ], [ 1, 6 ], [ -2, 0 ], [ 1, 2 ], [ 1, 2 ], [ 1, 2 ], 0, 0, 1, 0, 0, 0, 0 ] gap> # compute all vectors in the lattice gap> ContainedDecomposables( rat, SizesCentralizers( subtbl ), rest, > ReturnTrue ); [ [ 231, 7, -9, -9, 6, 15, 15, -1, -1, 1, 6, 6, 1, 1, -2, 1, 2, 2, 0, 0, 1, 0, 0, 0, 0 ], [ 231, 7, -9, -9, 6, 15, 15, 15, 15, 1, 6, 6, 1, 1, -2, 1, 2, 2, 0, 0, 1, 0, 0, 0, 0 ], [ 231, 7, -9, 7, 6, 15, 15, -1, -1, 1, 6, 6, 1, 1, -2, 1, 2, 2, 0, 0, 1, 0, 0, 0, 0 ], [ 231, 7, -9, 7, 6, 15, 15, 15, 15, 1, 6, 6, 1, 1, -2, 1, 2, 2, 0, 0, 1, 0, 0, 0, 0 ] ] gap> # compute only those vectors that are characters gap> ContainedDecomposables( rat, SizesCentralizers( subtbl ), rest, > x -> NonnegIntScalarProducts( subtbl, Irr( subtbl ), x ) ); [ [ 231, 7, -9, -9, 6, 15, 15, -1, -1, 1, 6, 6, 1, 1, -2, 1, 2, 2, 0, 0, 1, 0, 0, 0, 0 ], [ 231, 7, -9, 7, 6, 15, 15, -1, -1, 1, 6, 6, 1, 1, -2, 1, 2, 2, 0, 0, 1, 0, 0, 0, 0 ] ]
In the argument lists of the functions Congruences (73.6-2), ConsiderKernels (73.6-3), and ConsiderSmallerPowerMaps (73.6-4), tbl is an ordinary character table, chars a list of (values lists of) characters of tbl, prime a prime integer, approxmap a parametrized map that is an approximation for the prime-th power map of tbl (e.g., a list returned by InitPowerMap (73.6-1), and quick a Boolean.
The quick value true means that only those classes are considered for which approxmap lists more than one possible image.
‣ InitPowerMap( tbl, prime ) | ( function ) |
For an ordinary character table tbl and a prime prime, InitPowerMap returns a parametrized map that is a first approximation of the prime-th powermap of tbl, using the conditions 1. and 2. listed in the description of PossiblePowerMaps (73.1-2).
If there are classes for which no images are possible, according to these criteria, then fail is returned.
gap> t:= CharacterTable( "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 ] ]
‣ Congruences( tbl, chars, approxmap, prime, quick ) | ( function ) |
Congruences replaces the entries of approxmap by improved values, according to condition 3. listed in the description of PossiblePowerMaps (73.1-2).
For each class for which no images are possible according to the tests, the new value of approxmap is an empty list. Congruences returns true if no such inconsistencies occur, and false otherwise.
gap> Congruences( t, Irr( t ), pow, 2, false ); 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 ], 8, 9, 9, 10, 11, 12, 16, 16, 16, 16, 17, 17, 18, 18, [ 18, 19 ], [ 18, 19 ], 20, 20, 20, 20, 22, 22, 24, 24, [ 25, 26 ], [ 25, 26 ], 28, 28, 29, 29 ]
‣ ConsiderKernels( tbl, chars, approxmap, prime, quick ) | ( function ) |
ConsiderKernels replaces the entries of approxmap by improved values, according to condition 4. listed in the description of PossiblePowerMaps (73.1-2).
Congruences (73.6-2) returns true if the orders of the kernels of all characters in chars divide the order of the group of tbl, and false otherwise.
gap> t:= CharacterTable( "A7.2" );; init:= InitPowerMap( t, 2 ); [ 1, 1, 3, 4, [ 2, 9, 10 ], 6, 3, 8, 1, 1, [ 2, 9, 10 ], 3, [ 3, 4 ], 6, [ 7, 12 ] ] gap> ConsiderKernels( t, Irr( t ), init, 2, false ); true gap> init; [ 1, 1, 3, 4, 2, 6, 3, 8, 1, 1, 2, 3, [ 3, 4 ], 6, 7 ]
‣ ConsiderSmallerPowerMaps( tbl, approxmap, prime, quick ) | ( function ) |
ConsiderSmallerPowerMaps replaces the entries of approxmap by improved values, according to condition 5. listed in the description of PossiblePowerMaps (73.1-2).
ConsiderSmallerPowerMaps returns true if each class admits at least one image after the checks, otherwise false is returned. If no element orders of tbl are stored (see OrdersClassRepresentatives (71.9-1)) then true is returned without any tests.
gap> t:= CharacterTable( "3.A6" );; init:= InitPowerMap( t, 5 ); [ 1, [ 2, 3 ], [ 2, 3 ], 4, [ 5, 6 ], [ 5, 6 ], [ 7, 8 ], [ 7, 8 ], 9, [ 10, 11 ], [ 10, 11 ], 1, [ 2, 3 ], [ 2, 3 ], 1, [ 2, 3 ], [ 2, 3 ] ] gap> Indeterminateness( init ); 4096 gap> ConsiderSmallerPowerMaps( t, init, 5, false ); true gap> Indeterminateness( init ); 256
‣ MinusCharacter( character, primepowermap, prime ) | ( function ) |
Let character be (the list of values of) a class function χ, prime a prime integer p, and primepowermap a parametrized map that is an approximation of the p-th power map for the character table of χ. MinusCharacter returns the parametrized map of values of χ^{p-}, which is defined by χ^{p-}(g) = ( χ(g)^p - χ(g^p) ) / p.
gap> tbl:= CharacterTable( "S7" );; pow:= InitPowerMap( tbl, 2 );; gap> pow; [ 1, 1, 3, 4, [ 2, 9, 10 ], 6, 3, 8, 1, 1, [ 2, 9, 10 ], 3, [ 3, 4 ], 6, [ 7, 12 ] ] gap> chars:= Irr( tbl ){ [ 2 .. 5 ] };; gap> List( chars, x -> MinusCharacter( x, pow, 2 ) ); [ [ 0, 0, 0, 0, [ 0, 1 ], 0, 0, 0, 0, 0, [ 0, 1 ], 0, 0, 0, [ 0, 1 ] ] , [ 15, -1, 3, 0, [ -2, -1, 0 ], 0, -1, 1, 5, -3, [ 0, 1, 2 ], -1, 0, 0, [ 0, 1 ] ], [ 15, -1, 3, 0, [ -1, 0, 2 ], 0, -1, 1, 5, -3, [ 1, 2, 4 ], -1, 0, 0, 1 ], [ 190, -2, 1, 1, [ 0, 2 ], 0, 1, 1, -10, -10, [ 0, 2 ], -1, -1, 0, [ -1, 0 ] ] ]
‣ PowerMapsAllowedBySymmetrizations( tbl, subchars, chars, approxmap, prime, parameters ) | ( function ) |
Let tbl be an ordinary character table, prime a prime integer, approxmap a parametrized map that is an approximation of the prime-th power map of tbl (e.g., a list returned by InitPowerMap (73.6-1), chars and subchars two lists of (values lists of) characters of tbl, and parameters a record with components maxlen, minamb, maxamb (three integers), quick (a Boolean), and contained (a function). Usual values of contained are ContainedCharacters (73.5-17) or ContainedPossibleCharacters (73.5-15).
PowerMapsAllowedBySymmetrizations replaces the entries of approxmap by improved values, according to condition 6. listed in the description of PossiblePowerMaps (73.1-2).
More precisely, the strategy used is as follows.
First, for each χ ∈ chars, let minus:= MinusCharacter(χ, approxmap, prime).
If Indeterminateness( minus )= 1 and parameters.quick = false then the scalar products of minus with subchars are checked; if not all scalar products are nonnegative integers then an empty list is returned, otherwise χ is deleted from the list of characters to inspect.
Otherwise if Indeterminateness( minus ) is smaller than parameters.minamb then χ is deleted from the list of characters.
If parameters.minamb ≤ Indeterminateness( minus ) ≤ parameters.maxamb then construct the list of contained class functions poss:= parameters.contained(tbl, subchars, minus) and Parametrized( poss ), and improve the approximation of the power map using UpdateMap (73.5-7).
If this yields no further immediate improvements then we branch. If there is a character from chars left with less or equal parameters.maxlen possible symmetrizations, compute the union of power maps allowed by these possibilities. Otherwise we choose a class C such that the possible symmetrizations of a character in chars differ at C, and compute recursively the union of all allowed power maps with image at C fixed in the set given by the current approximation of the power map.
gap> tbl:= CharacterTable( "U4(3).4" );; gap> pow:= InitPowerMap( tbl, 2 );; gap> Congruences( tbl, Irr( tbl ), pow, 2 );; pow; [ 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 ], 8, 9, 9, 10, 11, 12, 16, 16, 16, 16, 17, 17, 18, 18, [ 18, 19 ], [ 18, 19 ], 20, 20, 20, 20, 22, 22, 24, 24, [ 25, 26 ], [ 25, 26 ], 28, 28, 29, 29 ] gap> PowerMapsAllowedBySymmetrizations( tbl, Irr( tbl ), Irr( tbl ), > 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 ] ]
‣ InitFusion( subtbl, tbl ) | ( function ) |
For two ordinary character tables subtbl and tbl, InitFusion returns a parametrized map that is a first approximation of the class fusion from subtbl to tbl, using condition 1. listed in the description of PossibleClassFusions (73.3-6).
If there are classes for which no images are possible, according to this criterion, then fail is returned.
gap> subtbl:= CharacterTable( "2F4(2)" );; tbl:= CharacterTable( "Ru" );; gap> fus:= InitFusion( subtbl, tbl ); [ 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 ] ]
‣ CheckPermChar( subtbl, tbl, approxmap, permchar ) | ( function ) |
CheckPermChar replaces the entries of the parametrized map approxmap by improved values, according to condition 3. listed in the description of PossibleClassFusions (73.3-6).
CheckPermChar returns true if no inconsistency occurred, and false otherwise.
gap> permchar:= Sum( Irr( tbl ){ [ 1, 5, 6 ] } );; gap> CheckPermChar( subtbl, tbl, fus, permchar ); fus; true [ 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 ]
‣ ConsiderTableAutomorphisms( approxmap, grp ) | ( function ) |
ConsiderTableAutomorphisms replaces the entries of the parametrized map approxmap by improved values, according to condition 4. listed in the description of PossibleClassFusions (73.3-6).
Afterwards exactly one representative of fusion maps (contained in approxmap) in each orbit under the action of the permutation group grp is contained in the modified parametrized map.
ConsiderTableAutomorphisms returns the list of positions where approxmap was changed.
gap> ConsiderTableAutomorphisms( fus, AutomorphismsOfTable( tbl ) ); [ 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 ]
‣ FusionsAllowedByRestrictions( subtbl, tbl, subchars, chars, approxmap, parameters ) | ( function ) |
Let subtbl and tbl be ordinary character tables, subchars and chars two lists of (values lists of) characters of subtbl and tbl, respectively, approxmap a parametrized map that is an approximation of the class fusion of subtbl in tbl, and parameters a record with the mandatory components maxlen, minamb, maxamb (three integers), quick (a Boolean), and contained (a function, usual values are ContainedCharacters (73.5-17) or ContainedPossibleCharacters (73.5-15)); optional components of the parameters record are testdec (the function that tests the decomposability, the default is NonnegIntScalarProducts (73.5-15)), powermaps (the power paps of subtbl that shall be used for compatibility checks, the default is the ComputedPowerMaps (73.1-1) value), subpowermaps (the power paps of tbl that shall be used for compatibility checks, the default is the ComputedPowerMaps (73.1-1) value).
FusionsAllowedByRestrictions replaces the entries of approxmap by improved values, according to condition 5. listed in the description of PossibleClassFusions (73.3-6).
More precisely, the strategy used is as follows.
First, for each χ ∈ chars, let restricted:= CompositionMaps( χ, approxmap ).
If Indeterminateness( restricted )= 1 and parameters.quick = false then the scalar products of restricted with subchars are checked; if not all scalar products are nonnegative integers then an empty list is returned, otherwise χ is deleted from the list of characters to inspect.
Otherwise if Indeterminateness( minus ) is smaller than parameters.minamb then χ is deleted from the list of characters.
If parameters.minamb ≤ Indeterminateness( restricted ) ≤ parameters.maxamb then construct poss:= parameters.contained( subtbl, subchars, restricted ) and Parametrized( poss ), and improve the approximation of the fusion map using UpdateMap (73.5-7).
If this yields no further immediate improvements then we branch. If there is a character from chars left with less or equal parameters.maxlen possible restrictions, compute the union of fusion maps allowed by these possibilities. Otherwise we choose a class C such that the possible restrictions of a character in chars differ at C, and compute recursively the union of all allowed fusion maps with image at C fixed in the set given by the current approximation of the fusion map.
gap> subtbl:= CharacterTable( "U3(3)" );; tbl:= CharacterTable( "J4" );; gap> fus:= InitFusion( subtbl, tbl );; gap> TestConsistencyMaps( ComputedPowerMaps( subtbl ), fus, > ComputedPowerMaps( tbl ) ); true gap> fus; [ 1, 2, 4, 4, [ 5, 6 ], [ 5, 6 ], [ 5, 6 ], 10, [ 12, 13 ], [ 12, 13 ], [ 14, 15, 16 ], [ 14, 15, 16 ], [ 21, 22 ], [ 21, 22 ] ] gap> ConsiderTableAutomorphisms( fus, AutomorphismsOfTable( tbl ) ); [ 9 ] gap> 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( subtbl, tbl, Irr( subtbl ), > Irr( tbl ), 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 ] ]
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