GAP (ref) - Chapter 73: Maps Concerning Character Tables

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73 Maps Concerning Character Tables
73.1 Power Maps
73.1-1 PowerMap
73.1-2 PossiblePowerMaps
73.1-3 ElementOrdersPowerMap
73.1-4 PowerMapByComposition
73.2 Orbits on Sets of Possible Power Maps
73.2-1 OrbitPowerMaps
73.2-2 RepresentativesPowerMaps
73.3 Class Fusions between Character Tables
73.3-1 FusionConjugacyClasses
73.3-2 ComputedClassFusions
73.3-3 GetFusionMap
73.3-4 StoreFusion
73.3-5 NamesOfFusionSources
73.3-6 PossibleClassFusions
73.3-7 ConsiderStructureConstants
73.4 Orbits on Sets of Possible Class Fusions
73.4-1 OrbitFusions
73.4-2 RepresentativesFusions
73.5 Parametrized Maps
73.5-1 CompositionMaps
73.5-2 InverseMap
73.5-3 ProjectionMap
73.5-4 Indirected
73.5-5 Parametrized
73.5-6 ContainedMaps
73.5-7 UpdateMap
73.5-8 MeetMaps
73.5-9 CommutativeDiagram
73.5-10 CheckFixedPoints
73.5-11 TransferDiagram
73.5-12 TestConsistencyMaps
73.5-13 Indeterminateness
73.5-14 PrintAmbiguity
73.5-15 ContainedSpecialVectors
73.5-16 CollapsedMat
73.5-17 ContainedDecomposables
73.6 Subroutines for the Construction of Power Maps
73.6-1 InitPowerMap
73.6-2 Congruences
73.6-3 ConsiderKernels
73.6-4 ConsiderSmallerPowerMaps
73.6-5 MinusCharacter
73.6-6 PowerMapsAllowedBySymmetrizations
73.7 Subroutines for the Construction of Class Fusions
73.7-1 InitFusion
73.7-2 CheckPermChar
73.7-3 ConsiderTableAutomorphisms
73.7-4 FusionsAllowedByRestrictions

### 73 Maps Concerning Character Tables

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

#### 73.1 Power Maps

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.

##### 73.1-1 PowerMap

`‣ 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

##### 73.1-2 PossiblePowerMaps

`‣ 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.

1. 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.

2. 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).

3. For each character \(\chi\) of \(G\) and each element \(g\) in \(G\), the values \(\chi(g^p)\) and `GaloisCyc`\(( \chi(g), p )\) 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 \(\lambda\) among these characters, the condition \(\chi(g)^p = \chi(g^p)\) is checked. The corresponding function is `Congruences` (73.6-2).

4. For each character \(\chi\) of \(G\), the kernel is a normal subgroup \(N\), and \(g^p \in N\) for all \(g \in N\); moreover, if \(N\) has index \(p\) in \(G\) then \(g^p \in N\) for all \(g \in G\), and if the index of \(N\) in \(G\) is coprime to \(p\) then \(g^p \not \in N\) for each \(g \not \in N\). These conditions are checked for the kernels of all characters \(\chi\) specified below, the corresponding function is `ConsiderKernels` (73.6-3).

5. If \(p\) is larger than the order \(m\) of an element \(g \in G\) 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).

6. For each character \(\chi\) of \(G\), the symmetrization \(\psi\) defined by \(\psi(g) = (\chi(g)^p - \chi(g^p))/p\) is a character. This condition is checked for the kernels of all characters \(\chi\) 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 ] ]

##### 73.1-3 ElementOrdersPowerMap

`‣ 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

##### 73.1-4 PowerMapByComposition

`‣ 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 ]

#### 73.2 Orbits on Sets of Possible Power Maps

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.

##### 73.2-1 OrbitPowerMaps

`‣ 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).

##### 73.2-2 RepresentativesPowerMaps

`‣ 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 ]

#### 73.3 Class Fusions between Character Tables

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.

##### 73.3-1 FusionConjugacyClasses

`‣ 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 ] ]

##### 73.3-2 ComputedClassFusions

`‣ 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.

`name`

  the `Identifier` (71.9-8) value of the character table to which the fusion maps,

`map`

  the 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).

##### 73.3-3 GetFusionMap

`‣ 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).

##### 73.3-4 StoreFusion

`‣ 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 ]

##### 73.3-5 NamesOfFusionSources

`‣ 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"

##### 73.3-6 PossibleClassFusions

`‣ 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.

1. 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).

2. The class fusion commutes with power maps. This is checked using `TestConsistencyMaps` (73.5-12).

3. 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).

4. 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.)

5. For each character \(\chi\) 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).

6. 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:

`chars`

  a list of characters of tbl which are used for the check of 5.; the default is `Irr( tbl )`,

`subchars`

  a list of characters of subtbl which are constituents of the restrictions of `chars`, the default is `Irr( subtbl )`,

`fusionmap`

  a parametrized map which is an approximation of the desired map,

`decompose`

  a 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,

`quick`

  a 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),

`verify`

  a 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),

`parameters`

  a 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 ] ]

##### 73.3-7 ConsiderStructureConstants

`‣ 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 \(\sigma\) in fusions with the property that for all triples \((i,j,k)\) of class positions, `ClassMultiplicationCoefficient`\(( \textit{subtbl}, i, j, k )\) is not bigger than `ClassMultiplicationCoefficient`\(( \textit{tbl}, \sigma[i], \sigma[j], \sigma[k] )\); 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.

#### 73.4 Orbits on Sets of Possible Class Fusions

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).

##### 73.4-1 OrbitFusions

`‣ 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.

##### 73.4-2 RepresentativesFusions

`‣ 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 ] ]

#### 73.5 Parametrized Maps

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.

##### 73.5-1 CompositionMaps

`‣ CompositionMaps`( paramap2, paramap1[, class] ) | ( function )

The composition of two parametrized maps paramap1, paramap2 is defined as the parametrized map comp that contains all compositions \(f_2 \circ f_1\) of elements \(f_1\) of paramap1 and \(f_2\) of paramap2. For example, the composition of a character \(\chi\) 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 \(\chi\) 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 ] ]

##### 73.5-2 InverseMap

`‣ 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 ] ]

##### 73.5-3 ProjectionMap

`‣ 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 ]

##### 73.5-4 Indirected

`‣ 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 ] ]

##### 73.5-5 Parametrized

`‣ 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 ] ]

##### 73.5-6 ContainedMaps

`‣ 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 ] ]

##### 73.5-7 UpdateMap

`‣ 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 ] ]

##### 73.5-8 MeetMaps

`‣ 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 ] ]

##### 73.5-9 CommutativeDiagram

`‣ 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`, \(\ldots\), `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 := [  ] )

##### 73.5-10 CheckFixedPoints

`‣ 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 ] ]

##### 73.5-11 TransferDiagram

`‣ 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 ]

##### 73.5-12 TestConsistencyMaps

`‣ 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

##### 73.5-13 Indeterminateness

`‣ 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

##### 73.5-14 PrintAmbiguity

`‣ 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 ]

##### 73.5-15 ContainedSpecialVectors

`‣ 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

##### 73.5-16 CollapsedMat

`‣ CollapsedMat`( mat, maps ) | ( function )

is a record with the components

`fusion`

  fusion that collapses those columns of mat that are equal in mat and also for all maps in the list maps,

`mat`

  the 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 ] ] )

##### 73.5-17 ContainedDecomposables

`‣ 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 \(\chi\) of parachar that satisfy func\(( \chi ) =\) `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 \(\{ \textit{moduls}[i] \cdot e_i; 1 \leq i \leq n \}\) 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 ] ]

#### 73.6 Subroutines for the Construction of Power Maps

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.

##### 73.6-1 InitPowerMap

`‣ 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 ] ]

##### 73.6-2 Congruences

`‣ 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 ]

##### 73.6-3 ConsiderKernels

`‣ 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 ]

##### 73.6-4 ConsiderSmallerPowerMaps

`‣ 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

##### 73.6-5 MinusCharacter

`‣ MinusCharacter`( character, primepowermap, prime ) | ( function )

Let character be (the list of values of) a class function \(\chi\), 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 \(\chi\). `MinusCharacter` returns the parametrized map of values of \(\chi^{{p-}}\), which is defined by \(\chi^{{p-}}(g) = ( \chi(g)^p - \chi(g^p) ) / p\).

    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 ] ] ]

##### 73.6-6 PowerMapsAllowedBySymmetrizations

`‣ 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 \(\chi \in \textit{chars}\), let `minus:= MinusCharacter(`\(\chi\)`, 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 \(\chi\) is deleted from the list of characters to inspect.

- Otherwise if `Indeterminateness( minus )` is smaller than `parameters.minamb` then \(\chi\) is deleted from the list of characters.

- If `parameters.minamb` \(\leq\) `Indeterminateness( minus )` \(\leq\) `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 ] ]

#### 73.7 Subroutines for the Construction of Class Fusions

##### 73.7-1 InitFusion

`‣ 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 ] ]

##### 73.7-2 CheckPermChar

`‣ 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 ]

##### 73.7-3 ConsiderTableAutomorphisms

`‣ 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 ]

##### 73.7-4 FusionsAllowedByRestrictions

`‣ 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 \(\chi \in \textit{chars}\), let `restricted:= CompositionMaps(` \(\chi\)`, 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 \(\chi\) is deleted from the list of characters to inspect.

- Otherwise if `Indeterminateness( minus )` is smaller than `parameters.minamb` then \(\chi\) is deleted from the list of characters.

- If `parameters.minamb` \(\leq\) `Indeterminateness( restricted )` \(\leq\) `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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