The functions described below, used in various parts of the CHEVIE package, are of a general nature and should really be included in other parts of the GAP3 library. We include them here for the moment for the commodity of the reader.
SymmetricDifference( S, T)
This function returns the symmetric difference of the sets S and T,
which can be written in GAP3 as
Difference(Union(x,y),IntersectionSet(x,y).
gap> SymmetricDifference([1,2],[2,3]);
[ 1, 3 ]
DifferenceMultiSet( l, s )
This function returns the difference of the multisets l and s. That is, l and s are lists which may contain several times the same item. The result is a list which is like l, excepted if an item occurs a times in s, the first a occurrences of this item in l have been deleted (all the occurrences if a is greater than the times it occurred in l).
gap> DifferenceMultiSet("ababcbadce","edbca");
"abbac"
Rotation(l, i)
This function returns l rotated i steps.
gap> l:=[1..5];;
gap> Rotation(l,1);
[ 2, 3, 4, 5, 1 ]
gap> Rotation(l,0);
[ 1, 2, 3, 4, 5 ]
gap> Rotation(l,-1);
[ 5, 1, 2, 3, 4 ]
Rotations(l)
This function returns the list of rotations of the list l.
gap> Rotations("abcd");
[ "abcd", "bcda", "cdab", "dabc" ]
gap> Rotations([1,0,1,0]);
[ [ 1, 0, 1, 0 ], [ 0, 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 0, 1, 0, 1 ] ]
Inherit(rec1,rec2[,fields])
This functions copies to the record rec1 all the fields of the record rec2. If an additional argument fields is given, it should be a list of field names, and then only the fields specified by fields are copied. The function returns the modified rec1.
gap> r:=rec(a:=1,b:=2);
rec(
a := 1,
b := 2 )
gap> s:=rec(c:=3,d:=4);
rec(
c := 3,
d := 4 )
gap> Inherit(r,s);
rec(
a := 1,
b := 2,
c := 3,
d := 4 )
gap> r:=rec(a:=1,b:=2);
rec(
a := 1,
b := 2 )
gap> Inherit(r,s,["d"]);
rec(
a := 1,
b := 2,
d := 4 )
Dictionary()
This function creates a dictionary data type. The created object is a record with two functions:
Get(k):k; it returns false if there
is no such key.
Insert(k,v) sets in the dictionary the value associated to key k to be
v.
The main advantage compared to records is that keys may be of any type.
gap> d:=Dictionary();
Dictionary with 0 entries
gap> d.Insert("a",1);
1
gap> d.Insert("b",2);
2
gap> d.Get("a");
1
gap> d.Get("c");
false
gap> d;
Dictionary with 2 entries
GetRoot( x, n [, msg])
n must be a positive integer. GetRoot returns an n-th root of
x when possible, else signals an error. If msg is present and
InfoChevie=Print a warning message is printed about which choice of
root has been made, after printing msg.
In the current implementation, it is possible to find an n-th root when x is one of the following GAP3 objects:
1- a monomial of the form a*q^(b*n) when we know how to find a root
of a. The root chosen is GetRoot(a,n)*q^b.
2- a root of unity of the form E(a)^i. The root chosen is
E(a*n)^i.
3- an integer, when n=2 (the root chosen is ER(x)) or when x is a
perfect n-th power of a (the root chosen is a).
4- a product of an x of form 2- by an x of form 3-.
5- when x is a record and has a method x.operations.GetRoot the work
is delegated to that method.
gap> q:=X(Cyclotomics);;q.name:="q";;
gap> GetRoot(E(3)*q^2,2,"test");
#warning: test: E3^2q chosen as 2nd root of (E(3))*q^2
(E(3)^2)*q
gap> GetRoot(1,2,"test");
#warning: test: 1 chosen as 2nd root of 1
1
The example above shows that GetRoot is not compatible with
specialization: E(3)*q^2 evaluated at E(3) is 1 whose root
chosen by GetRoot is 1, while (-E(3)^2)*q evaluated at E(3) is
-1. Actually it can be shown that it is not possible mathematically to
define a function GetRoot compatible with specializations. This is why
there is a provision in functions for character tables of Hecke algebras
to provide explicit roots.
gap> GetRoot(8,3);
2
gap> GetRoot(7,3);
Error, : unable to compute 3-th root of 7
in
GetRoot( 7, 3 ) called from
main loop
brk>
CharParams(G)
G should be a group or another object which has a method
CharTable, or a character table. The function CharParams tries to
determine a list of labels for the characters of G. If G has a
method CharParams this is called. Otherwise, if G is not a
character table, its CharTable is called. If the table has a field
.charparam in .irredinfo this is returned. Otherwise, the list
[1..Length(G.irreducibles)] is returned.
gap> CharParams(CoxeterGroup("A",2));
[ [ [ 1, 1, 1 ] ], [ [ 2, 1 ] ], [ [ 3 ] ] ]
gap> CharParams(Group((1,2),(2,3)));
#W Warning: Group has no name
[ 1 .. 3 ]
CharName(G, param)
G should be a group and param a parameter of a character of that
group (as returned by CharParams). If G has a method CharName, the
function returns the result of that method, which is a string which
displays nicely param (this is used by CHEVIE to nicely fill the
.charNames in a CharTable -- all finite reflection groups have such
methods CharName).
gap> G:=CoxeterGroup("G", 2);
CoxeterGroup("G",2)
gap> CharParams(G);
[ [ [ 1, 0 ] ], [ [ 1, 6 ] ], [ [ 1, 3, 1 ] ], [ [ 1, 3, 2 ] ],
[ [ 2, 1 ] ], [ [ 2, 2 ] ] ]
gap> List(last,x->CharName(G,x));
[ "phi{1,0}", "phi{1,6}", "phi{1,3}'", "phi{1,3}''", "phi{2,1}",
"phi{2,2}" ]
PositionId( G )
G should be a group, a character table, an Hecke algebra or another
object which has characters. PositionId returns the position of the
identity character in the character table of G.
gap> W := CoxeterGroup( "D", 4 );;
gap> PositionId( W );
13
PermutationOnClasses( G, aut)
aut is an automorphism of the group G (for a permutation group, this
could be given as a permutation normalizing G). The result is the
permutation of [1..NrConjugacyClasses(G)] induced ny aut.
gap> WF:=RootDatum("3D4");
3D4
gap> PermutationOnClasses(Group(WF),WF.phi);
( 2, 8, 7)( 5,13,12)
103.12 PermutationOnCharacters
PermutationOnCharacters( G, aut)
aut is an automorphism of the group G (for a permutation group, this
could be given as a permutation normalizing G). The result is the
permutation of [1..NrConjugacyClasses(G)] (representing indices of
irreducible characters) induced by aut.
gap> WF:=RootDatum("3D4");
3D4
gap> PermutationOnCharacters(Group(WF),WF.phi);
( 1, 7, 2)( 8,12, 9)
InductionTable( S, G )
InductionTable computes the decomposition of the induced characters from
the subgroup S into irreducible characters of G. The rows correspond to
the characters of the parent group, the columns to those of the subgroup.
What is returned is actually a record with several fields: .scalar
contains the induction table proper, and there are Display and Format
methods. The other fields contain labeling information taken from the
character tables of S and G when it exists.
gap> G := Group( [ (1,2), (2,3), (3,4) ], () );
Group( (1,2), (2,3), (3,4) )
gap> S:=Subgroup( G, [ (1,2), (3,4) ] );
Subgroup( Group( (1,2), (2,3), (3,4) ), [ (1,2), (3,4) ] )
gap> G.name := "G";; S.name := "S";; # to avoid warnings
gap> Display( InductionTable( S, G ) );
Induction from S to G
|X.1 X.2 X.3 X.4
_____________________
X.1 | 1 . . .
X.2 | . . . 1
X.3 | 1 . . 1
X.4 | . 1 1 1
X.5 | 1 1 1 .
gap> G := CoxeterGroup( "G", 2 );;
gap> S := ReflectionSubgroup( G, [ 1, 4 ] );
ReflectionSubgroup(CoxeterGroup("G",2), [ 1, 4 ])
gap> t := InductionTable( S, G );
InductionTable(ReflectionSubgroup(CoxeterGroup("G",2), [ 1, 4 ]), Coxe\
terGroup("G",2))
gap> Display( t );
Induction from A1x~A1 to G2
|11,11 11,2 2,11 2,2
________________________________
phi{1,0} | . . . 1
phi{1,6} | 1 . . .
phi{1,3}' | . 1 . .
phi{1,3}'' | . . 1 .
phi{2,1} | . 1 1 .
phi{2,2} | 1 . . 1
The Display and Format methods take the same arguments as the
FormatTable method. For instance to select a subset of the characters of
the subgroup and of the parent group, one can call
gap> Display( t,rec( rows := [5], columns := [3,2] ) );
Induction from A1x~A1 to G2
|2,11 11,2
____________________
phi{2,1} | 1 1
It is also possible to get TeX and LaTeX output, see FormatTable.
103.14 CharRepresentationWords
CharRepresentationWords( rep , elts )
given a list rep of matrices corresponding to generators and a list elts of words in the generators it returns the list of traces of the corresponding representation on the elements in elts.
gap> H := Hecke(CoxeterGroup( "F", 4 ));;
gap> r := ChevieClassInfo( Group( H ) ).classtext;;
gap> t := HeckeReflectionRepresentation( H );;
gap> CharRepresentationWords( t, r );
[ 4, -4, 0, 1, -1, 0, 1, -1, -2, 2, 0, 2, -2, -1, 1, 0, 2, -2, -1, 1,
0, 0, 2, -2, 0 ]
Transversals( G[, m] )
returns a pair [orb, rep] where orb is a list of the orbits of the
permutation group G on [ 1..LargestMovedPoint( G ) ] and rep is a
list of list of elements of G such that rep[i][j] applied to
orb[i][1] yields orb[i][j] for all i,j. If the optional argument
m is given, then LargestMovedPoint( G ) is replaced by the integer
m.
gap> G := Group( (1,7)(2,3)(5,6)(8,9)(11,12),
> (1,5)(2,8)(3,4)(7,11)(9,10) );;
gap> Transversals( G );
[ [ [ 1, 7, 5, 11, 6, 12 ], [ 2, 3, 8, 4, 9, 10 ] ],
[ [ (), ( 1, 7)( 2, 3)( 5, 6)( 8, 9)(11,12),
( 1, 5)( 2, 8)( 3, 4)( 7,11)( 9,10),
( 1,11,12, 7, 5, 6)( 2, 4, 3, 8,10, 9),
( 1, 6, 5, 7,12,11)( 2, 9,10, 8, 3, 4),
( 1,12)( 2, 4)( 3, 9)( 6, 7)( 8,10) ],
[ (), ( 1, 7)( 2, 3)( 5, 6)( 8, 9)(11,12),
( 1, 5)( 2, 8)( 3, 4)( 7,11)( 9,10),
( 1,11,12, 7, 5, 6)( 2, 4, 3, 8,10, 9),
( 1, 6, 5, 7,12,11)( 2, 9,10, 8, 3, 4),
( 1, 6)( 2,10)( 4, 8)( 5,11)( 7,12) ] ] ]
AbelianGenerators( A)
A should be an abelian group or the list of its generators. The function
returns a list of generators for A of orders given by
AbelianInvariants(A).
gap3-jm