Goto Chapter: Top 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 Bib Ind

73 Maps Concerning Character Tables

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

73.5-1 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

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 \(\chi\) 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).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).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).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(`

is known, otherwise`tbl`)`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.

`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).

`‣ 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( `

, and returns the `destination` )`map`

component of the first such record. `GetFusionMap( `

fetches that fusion map for which the record additionally has the `source`, `destination`, `specification` )`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 \(\chi\) 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).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(`

is known, otherwise`subtbl`)`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`

,`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 ] ]

`‣ 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 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

or `map`[i] = `paramap`[i]

is contained in the list `map`[i]

; we say then that `paramap`[i]`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 \(f_2 \circ f_1\) 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(`

is a parametrized map with entry `paramap2`, `paramap1`)`CompositionMaps(`

at position `paramap2`, `paramap1`, `class`)`class`. If

is an integer then `paramap1`[`class`]`CompositionMaps(`

is equal to `paramap2`, `paramap1`, `class`)

. Otherwise it is the union of `paramap2`[ `paramap1`[ `class` ] ]

for `paramap2`[`i`]`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

exactly for \(j\), and equal to the set of all preimages of \(i\) under `paramap`[`j`]`paramap` otherwise.

We have `CompositionMaps( `

the identity map.`paramap`, InverseMap( `paramap` ) )

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( `

the identity map, i.e., first projecting and then fusing yields the identity. Note that `fusionmap`, ProjectionMap( `fusionmap` ) )`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( `

is equal to `paramap` ) )`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`

, \(\ldots\), `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 := [ ] )

`‣ 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

`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 ] ] )

`‣ 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 ] ]

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 \(\chi\), `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 ] ] ]

`‣ 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

then the scalar products of`parameters`.quick = false`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

then \(\chi\) is deleted from the list of characters.`parameters`.minambIf

\(\leq\)`parameters`.minamb`Indeterminateness( minus )`

\(\leq\)

then construct the list of contained class functions`parameters`.maxamb`poss:=`

and`parameters`.contained(`tbl`,`subchars`, minus)`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

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 `parameters`.maxlen`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 \(\chi \in \textit{chars}\), let `restricted:= CompositionMaps( `

\(\chi\)`, `

.`approxmap` )

If

`Indeterminateness( restricted )`

\( = 1\) and

then the scalar products of`parameters`.quick = false`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

then \(\chi\) is deleted from the list of characters.`parameters`.minambIf

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

\(\leq\)

then construct`parameters`.maxamb`poss:=`

and`parameters`.contained(`subtbl`,`subchars`, restricted )`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 ] ]

Goto Chapter: Top 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 Bib Ind

generated by GAPDoc2HTML