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 GAP library. We include them here for the moment for the commodity of the reader.
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 ]
This function requires the package "chevie" (see RequirePackage).
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, 3, "''" ] ], [ [ 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}" ]
This function requires the package "chevie" (see RequirePackage).
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
This function requires the package "chevie" (see RequirePackage).
90.4 InductionTable
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 is a
Display
method. 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 | X1a X1b X1c X1d _____________________ X1a | 1 . . . X1b | . . . 1 X2a | 1 . . 1 X3a | . 1 1 1 X3b | 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 ]), CoxeterGroup("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
If one does not want to see the whole induction table, one can specify
the characters of the subgroup and of the parent group by giving a second
argument to Display
. This second argument is a record with optional
components charsGroup
and charsSubgroup
, to which one has to assign
the lists of rows and columns which should be printed.
gap> Display( t,rec( charsGroup := [5], charsSubgroup := [2,3] ) ); Induction from A1x~A1 to G2 | 11,2 2,11 ____________________ phi{2,1} | 1 1
This function requires the package "chevie" (see RequirePackage).
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 ]
This function requires the package "chevie" (see RequirePackage).
PositionClass( G, c )
G must be a domain for which ConjugacyClasses
is defined and c must
be an element of G. This functions returns a positive integer i
such
that c in ConjugacyClasses( G )[i]
.
gap> G := Group( (1,2)(3,4), (1,2,3,4,5) );; gap> ConjugacyClasses( G ); [ ConjugacyClass( Group( (1,2)(3,4), (1,2,3,4,5) ), () ), ConjugacyClass( Group( (1,2)(3,4), (1,2,3,4,5) ), (3,4,5) ), ConjugacyClass( Group( (1,2)(3,4), (1,2,3,4,5) ), (2,3)(4,5) ), ConjugacyClass( Group( (1,2)(3,4), (1,2,3,4,5) ), (1,2,3,4,5) ), ConjugacyClass( Group( (1,2)(3,4), (1,2,3,4,5) ), (1,2,3,5,4) ) ] gap> g := Random( G ); (1,2,5,4,3) gap> PositionClass( G, g ); 5
This function requires the package "chevie" (see RequirePackage).
90.7 PointsAndRepresentativesOrbits
PointsAndRepresentativesOrbits( 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> PointsAndRepresentativesOrbits( 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) ] ] ]
This function requires the package "chevie" (see RequirePackage).
SublistUnbnd( l, ind )
Sublist of a list with possibly unbound entries.
The writing of this function was prompted by the fact that if l has
some unbound entries, l{ind}
returns an error message instead of
doing what is expected (which is what this routine does).
gap> l := [ 1, , 2, , , 3 ];; gap> SublistUnbnd( l, [ 1..4 ] ); [ 1,, 2 ]
If you use l{[ 1..4 ]}
, you get an error message.
This function requires the package "chevie" (see RequirePackage).
Coefficient( a, b )
generic
routine which looks if a has a Coefficient
method in its
operations record and then returns a.operations.Coefficient(a,b)
.
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 GAP 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>
This function requires the package "chevie" (see RequirePackage).
IntListToString( part [, brackets] )
part must be a list of positive integers. If all of them are smaller
than 10 then a string of digits corresponding to the entries of part is
returned. If an entry is ≥ 10 then the elements of part are
converted to strings, concatenated with separating commas and the result
surrounded by brackets. By default ()
brackets are used. This may be
changed by giving as second argument a length two string specifying
another kind of brackets.
gap> IntListToString( [ 4, 2, 2, 1, 1 ] ); "42211" gap> IntListToString( [ 14, 2, 2, 1, 1 ] ); "(14,2,2,1,1)" gap> IntListToString( [ 14, 2, 2, 1, 1 ], "{}" ); "{14,2,2,1,1}"
This function requires the package "chevie" (see RequirePackage).
Join( list [, delimiter] )
This function is similar to the Perl function of the same name. It first
applies the function String
to all elements of the list, then joins
the resulting strings, separated by the given delimiter (if omitted,
","
is used as a delimiter)
gap> Join([1..4]); "1,2,3,4" gap> Join([1..4],"foo"); "1foo2foo3foo4"
This function requires the package "chevie" (see RequirePackage).
Split( s [, delimiter] )
This function is similar to the Perl function of the same name. It splits
the string s at each occurrence of the delimiter (a character).
If delimiter is omitted, ','
is used as a delimiter.
gap> Split("14,2,2,1,"); [ "14" , "2" , "2" , "1" , "" ]
This function requires the package "chevie" (see RequirePackage).
Replace( s [, s1, r1 [, s2, r2 [...]]])
Replaces in list s all (non-overlapping) occurrences of sublist s1 by list r1, then all occurrences of s2 by r2, etc...
gap> Replace("aabaabaabbb","aabaa","c","cba","def","bbb","ult"); "default"
This function requires the package "chevie" (see RequirePackage).
SymmetricDifference( S, T)
This function returns the symmetric difference of the sets S and T,
that is, Difference(Union(x,y),IntersectionSet(x,y)
.
gap>SymmetricDifference([1,2],[2,3]); [1,3]
This function requires the package "chevie" (see RequirePackage).
GAP 3.4.4