defcolonchar58
This section contains some examples of the use of GAP in character theory. First a few very simple commands for handling character tables are introduced, and afterwards we will construct the character tables of (A_5times 3)!colon!2 and of A_6.2^2.
GAP has a large library of character tables, so let us look at one of these tables, e.g., the table of the Mathieu group M_{11}:
gap> m11:= CharTable( "M11" );
CharTable( "M11" )
Character tables contain a lot of information. This is not printed in full length since the internal structure is not easy to read. The next statement shows a more comfortable output format.
gap> DisplayCharTable( m11 );
M11
2 4 4 1 3 . 1 3 3 . .
3 2 1 2 . . 1 . . . .
5 1 . . . 1 . . . . .
11 1 . . . . . . . 1 1
1a 2a 3a 4a 5a 6a 8a 8b 11a 11b
2P 1a 1a 3a 2a 5a 3a 4a 4a 11b 11a
3P 1a 2a 1a 4a 5a 2a 8a 8b 11a 11b
5P 1a 2a 3a 4a 1a 6a 8b 8a 11a 11b
11P 1a 2a 3a 4a 5a 6a 8a 8b 1a 1a
X.1 1 1 1 1 1 1 1 1 1 1
X.2 10 2 1 2 . -1 . . -1 -1
X.3 10 -2 1 . . 1 A -A -1 -1
X.4 10 -2 1 . . 1 -A A -1 -1
X.5 11 3 2 -1 1 . -1 -1 . .
X.6 16 . -2 . 1 . . . B /B
X.7 16 . -2 . 1 . . . /B B
X.8 44 4 -1 . -1 1 . . . .
X.9 45 -3 . 1 . . -1 -1 1 1
X.10 55 -1 1 -1 . -1 1 1 . .
A = E(8)+E(8)^3
= ER(-2) = i2
B = E(11)+E(11)^3+E(11)^4+E(11)^5+E(11)^9
= (-1+ER(-11))/2 = b11
We are not too much interested in the internal structure of this
character table (see Character Table Records); but of course we can
access all information about the centralizer orders (first four lines),
element orders (next line), power maps for the prime divisors of the
group order (next four lines), irreducible characters (lines parametrized
by X.1 ldots X.10) and irrational character values (last four
lines), see DisplayCharTable for a detailed description of the format
of the displayed table. E.g., the irreducible characters are a list with
name m11.irreducibles, and each character is a list of cyclotomic
integers (see chapter Cyclotomics). There are various ways to describe
the irrationalities; e.g., the square root of -2 can be entered as
E(8) + E(8)^3 or ER(-2), the famous ATLAS of Finite
Groups~CCN85 denotes it as i2.
gap> m11.irreducibles[3];
[ 10, -2, 1, 0, 0, 1, E(8)+E(8)^3, -E(8)-E(8)^3, -1, -1 ]
We can for instance form tensor products of this character with all irreducibles, and compute the decomposition into irreducibles.
gap> tens:= Tensored( [ last ], m11.irreducibles );;
gap> MatScalarProducts( m11, m11.irreducibles, tens );
[ [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 1 ],
[ 0, 0, 0, 0, 1, 0, 0, 1, 1, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 1, 0, 1 ],
[ 0, 0, 0, 1, 0, 0, 0, 0, 1, 1 ], [ 0, 0, 0, 0, 0, 0, 1, 1, 1, 1 ],
[ 0, 0, 0, 0, 0, 1, 0, 1, 1, 1 ], [ 0, 0, 1, 1, 0, 1, 1, 2, 3, 3 ],
[ 0, 1, 0, 1, 1, 1, 1, 3, 2, 3 ], [ 0, 1, 1, 0, 1, 1, 1, 3, 3, 4 ] ]
The decomposition means for example that the third character in the list
tens is the sum of the irreducible characters at positions 5, 8 and 9.
gap> tens[3];
[ 100, 4, 1, 0, 0, 1, -2, -2, 1, 1 ]
gap> tens[3] = Sum( Sublist( m11.irreducibles, [ 5, 8, 9 ] ) );
true
Or we can compute symmetrizations, e.g., the characters chi^{2+}, defined by chi^{2+}(g) = frac{1}{2} ( chi^2(g) + chi(g^2) ), for all irreducibles.
gap> sym:= SymmetricParts( m11, m11.irreducibles, 2 );;
gap> MatScalarProducts( m11, m11.irreducibles, sym );
[ [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 1, 1, 0, 0, 0, 0, 0, 1, 0, 0 ],
[ 0, 0, 0, 0, 1, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0, 1, 0, 0 ],
[ 1, 1, 0, 0, 1, 0, 0, 1, 0, 0 ], [ 0, 1, 0, 0, 1, 0, 1, 1, 0, 1 ],
[ 0, 1, 0, 0, 1, 1, 0, 1, 0, 1 ], [ 1, 3, 0, 0, 3, 2, 2, 8, 4, 6 ],
[ 1, 2, 0, 0, 3, 2, 2, 8, 4, 7 ],
[ 1, 3, 1, 1, 4, 3, 3, 11, 7, 10 ] ]
gap> sym[2];
[ 55, 7, 1, 3, 0, 1, 1, 1, 0, 0 ]
gap> sym[2] = Sum( Sublist( m11.irreducibles, [ 1, 2, 8 ] ) );
true
If the subgroup fusion into a supergroup is known, characters can be induced to this group, e.g., to obtain the permutation character of the action of M_{12} on the cosets of M_{11}.
gap> m12:= CharTable( "M12" );;
gap> permchar:= Induced( m11, m12, [ m11.irreducibles[1] ] );
[ [ 12, 0, 4, 3, 0, 0, 4, 2, 0, 1, 0, 2, 0, 1, 1 ] ]
gap> MatScalarProducts( m12, m12.irreducibles, last );
[ [ 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ]
gap> DisplayCharTable( m12, rec( chars:= permchar ) );
M12
2 6 4 6 1 2 5 5 1 2 1 3 3 1 . .
3 3 1 1 3 2 . . . 1 1 . . . . .
5 1 1 . . . . . 1 . . . . 1 . .
11 1 . . . . . . . . . . . . 1 1
1a 2a 2b 3a 3b 4a 4b 5a 6a 6b 8a 8b 10a 11a 11b
2P 1a 1a 1a 3a 3b 2b 2b 5a 3b 3a 4a 4b 5a 11b 11a
3P 1a 2a 2b 1a 1a 4a 4b 5a 2a 2b 8a 8b 10a 11a 11b
5P 1a 2a 2b 3a 3b 4a 4b 1a 6a 6b 8a 8b 2a 11a 11b
11P 1a 2a 2b 3a 3b 4a 4b 5a 6a 6b 8a 8b 10a 1a 1a
Y.1 12 . 4 3 . . 4 2 . 1 . 2 . 1 1
It should be emphasized that the heart of character theory is dealing
with lists. Characters are lists, and also the maps which occur are
represented as lists. Note that the multiplication of group elements is
not available, so we neither have homomorphisms. All we can talk of are
class functions, and the lists are regarded as such functions, being the
lists of images with respect to a fixed order of conjugacy classes.
Therefore we do not write chi( cl ) or cl^chi for the value of the
character chi on the class cl, but chi[i] where i is the position
of the class cl.
Since the data structures are so basic, most calculations involve compositions of maps; for example, the embedding of a subgroup in a group is described by the so--called subgroup fusion which is a class function that maps each class c of the subgroup to that class of the group that contains c. Consider the symmetric group S_5 cong A_5.2 as subgroup of M_{11}. (Do not worry about the names that are used to get library tables, see CharTable for an overview.)
gap> s5:= CharTable( "A5.2" );;
gap> map:= GetFusionMap( s5, m11 );
[ 1, 2, 3, 5, 2, 4, 6 ]
The subgroup fusion is already stored on the table. We see that class 1
of s5 is mapped to class 1 of m11 (which means that the identity of
S_5 maps to the identity of M_{11}), classes 2 and 5 of s5 both map
to class 2 of m11 (which means that all involutions of S_5 are
conjugate in M_{11}), and so on.
The restriction of a character of m11 to s5 is just the composition
of this character with the subgroup fusion map. Viewing this map as list
one would call this composition an indirection.
gap> chi:= m11.irreducibles[3];
[ 10, -2, 1, 0, 0, 1, E(8)+E(8)^3, -E(8)-E(8)^3, -1, -1 ]
gap> rest:= List( map, x -> chi[x] );
[ 10, -2, 1, 0, -2, 0, 1 ]
This looks very easy, and many GAP functions in character theory do such simple calculations. But note that it is not always obvious that a list is regarded as a map, where preimages and/or images refer to positions of certain conjugacy classes.
gap> alt:= s5.irreducibles[2];
[ 1, 1, 1, 1, -1, -1, -1 ]
gap> kernel:= KernelChar( last );
[ 1, 2, 3, 4 ]
The kernel of a character is represented as the list of (positions of)
classes lying in the kernel. We know that the kernel of the alternating
character alt of s5 is the alternating group A_5. The order of the
kernel can be computed as sum of the lengths of the contained classes
from the character table, using that the classlengths are stored in the
classes component of the table.
gap> s5.classes;
[ 1, 15, 20, 24, 10, 30, 20 ]
gap> last{ kernel };
[ 1, 15, 20, 24 ]
gap> Sum( last );
60
We chose those classlengths of s5 that belong to the S_5--classes
contained in the alternating group. The same thing is done in the
following command, reflecting the view of the kernel as map.
gap> List( kernel, x -> s5.classes[x] );
[ 1, 15, 20, 24 ]
gap> Sum( kernel, x -> s5.classes[x] );
60
This small example shows how the functions List and Sum can be used.
About Further List Operations, and we will make heavy use of them; in many cases such
a command might look very strange, but it is just the translation of a
(hardly less complicated) mathematical formula to character theory.
And now let us construct some small character tables!
setlengthunitlength0.1cm The group G = (A_5times 3)!colon!2 is a maximal subgroup of the alternating group A_8; G extends to S_5times S_3 in S_8. We want to construct the character table of G.
First the tables of the subgroup A_5times 3 and the supergroup S_5times S_3 are constructed; the tables of the factors of each direct product are again got from the table library using admissible names, see CharTable for this. put(10,-37)beginpicture(50,45) put(25,5)circle1 put(15,15)circle1 put(25,25)circle1 put(25,30)circle1 put(25,35)circle1 put(35,15)circle1 put(10,20)circle1 put(40,20)circle1 put(20,30)circle1 put(30,30)circle1 put(7,20)makebox(0,0)S_5 put(12,15)makebox(0,0)A_5 put(43,20)makebox(0,0)S_3 put(38,15)makebox(0,0)3 put(27,30)makebox(0,0)G put(25,37)makebox(0,0)S_5times S_3 put(14,30)makebox(0,0)S_5times 3 put(38,30)makebox(0,0)A_5times S_3 put(25,5)line(-1,1)15 put(25,5)line(1,1)15 put(35,15)line(-1,1)15 put(15,15)line(1,1)15 put(40,20)line(-1,1)15 put(10,20)line(1,1)15 put(25,25)line(0,1)10
gap> a5:= CharTable( "A5" );;
gap> c3:= CharTable( "Cyclic", 3 );;
gap> a5xc3:= CharTableDirectProduct( a5, c3 );;
gap> s5:= CharTable( "A5.2" );;
gap> s3:= CharTable( "Symmetric", 3 );;
gap> s3.irreducibles;
[ [ 1, -1, 1 ], [ 2, 0, -1 ], [ 1, 1, 1 ] ]
# The trivial character shall be the first one.
gap> SortCharactersCharTable( s3 ); # returns the applied permutation
(1,2,3)
gap> s5xs3:= CharTableDirectProduct( s5, s3 );;
G is the normal subgroup of index 2 in S_5times S_3 which contains
neither S_5 nor the normal S_3. We want to find the classes of
s5xs3 whose union is G. For that, we compute the set of kernels of
irreducibles --remember that they are given simply by lists of numbers of
contained classes-- and then choose those kernels belonging to normal
subgroups of index 2.
gap> kernels:= Set( List( s5xs3.irreducibles, KernelChar ) );
[ [ 1 ], [ 1, 2, 3 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ],
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
19, 20, 21 ], [ 1, 3 ],
[ 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21 ],
[ 1, 3, 4, 6, 7, 9, 10, 12, 14, 17, 20 ], [ 1, 4, 7, 10 ],
[ 1, 4, 7, 10, 13, 16, 19 ] ]
gap> sizes:= List( kernels, x -> Sum( Sublist( s5xs3.classes, x ) ) );
[ 1, 6, 360, 720, 3, 360, 360, 60, 120 ]
gap> s5xs3.size;
720
gap> index2:= Sublist( kernels, [ 3, 6, 7 ] );
[ [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ],
[ 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21 ],
[ 1, 3, 4, 6, 7, 9, 10, 12, 14, 17, 20 ] ]
In order to decide which kernel describes G, we consider the embeddings
of s5 and s3 in s5xs3, given by the subgroup fusions.
gap> s5ins5xs3:= GetFusionMap( s5, s5xs3 );
[ 1, 4, 7, 10, 13, 16, 19 ]
gap> s3ins5xs3:= GetFusionMap( s3, s5xs3 );
[ 1, 2, 3 ]
gap> Filtered( index2, x->Intersection(x,s5ins5xs3)<>s5ins5xs3 and
> Intersection(x,s3ins5xs3)<>s3ins5xs3 );
[ [ 1, 3, 4, 6, 7, 9, 10, 12, 14, 17, 20 ] ]
gap> nsg:= last[1];
[ 1, 3, 4, 6, 7, 9, 10, 12, 14, 17, 20 ]
We now construct a first approximation of the character table of this
normal subgroup, namely the restriction of s5xs3 to the classes given
by nsg.
gap> sub:= CharTableNormalSubgroup( s5xs3, nsg );;
#I CharTableNormalSubgroup: classes in [ 8 ] necessarily split
gap> PrintCharTable( sub );
rec( identifier := "Rest(A5.2xS3,[ 1, 3, 4, 6, 7, 9, 10, 12, 14, 17, 2\
0 ])", size :=
360, name := "Rest(A5.2xS3,[ 1, 3, 4, 6, 7, 9, 10, 12, 14, 17, 20 ])",\
order := 360, centralizers := [ 360, 180, 24, 12, 18, 9, 15, 15/2,
12, 4, 6 ], orders := [ 1, 3, 2, 6, 3, 3, 5, 15, 2, 4, 6
], powermap := [ , [ 1, 2, 1, 2, 5, 6, 7, 8, 1, 3, 5 ],
[ 1, 1, 3, 3, 1, 1, 7, 7, 9, 10, 9 ],,
[ 1, 2, 3, 4, 5, 6, 1, 2, 9, 10, 11 ] ], classes :=
[ 1, 2, 15, 30, 20, 40, 24, 48, 30, 90, 60
], operations := CharTableOps, irreducibles :=
[ [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ],
[ 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1 ],
[ 2, -1, 2, -1, 2, -1, 2, -1, 0, 0, 0 ],
[ 6, 6, -2, -2, 0, 0, 1, 1, 0, 0, 0 ],
[ 4, 4, 0, 0, 1, 1, -1, -1, 2, 0, -1 ],
[ 4, 4, 0, 0, 1, 1, -1, -1, -2, 0, 1 ],
[ 8, -4, 0, 0, 2, -1, -2, 1, 0, 0, 0 ],
[ 5, 5, 1, 1, -1, -1, 0, 0, 1, -1, 1 ],
[ 5, 5, 1, 1, -1, -1, 0, 0, -1, 1, -1 ],
[ 10, -5, 2, -1, -2, 1, 0, 0, 0, 0, 0 ] ], fusions := [ rec(
name := [ 'A', '5', '.', '2', 'x', 'S', '3' ],
map := [ 1, 3, 4, 6, 7, 9, 10, 12, 14, 17, 20 ] ) ] )
Not all restrictions of irreducible characters of s5xs3 to sub remain
irreducible. We compute those restrictions with norm larger than 1.
gap> red:= Filtered( Restricted( s5xs3, sub, s5xs3.irreducibles ),
> x -> ScalarProduct( sub, x, x ) > 1 );
[ [ 12, -6, -4, 2, 0, 0, 2, -1, 0, 0, 0 ] ]
gap> Filtered( [ 1 .. Length( nsg ) ],
> x -> not IsInt( sub.centralizers[x] ) );
[ 8 ]
Note that sub is not actually a character table in the sense of
mathematics but only a record with components like a character table.
GAP does not know about this subtleties and treats it as a character
table.
As the list centralizers of centralizer orders shows, at least class 8
splits into two conjugacy classes in G, since this is the only
possibility to achieve integral centralizer orders.
Since 10 restrictions of irreducible characters remain irreducible for
G (sub contains 10 irreducibles), only one of the 11 irreducibles of
S_5times S_3 splits into two irreducibles of G, in other words,
class 8 is the only splitting class.
Thus we create a new approximation of the desired character table (which
we call split) where this class is split; 8th and 9th column of the
known irreducibles are of course equal, and due to the splitting the
second powermap for these columns is ambiguous.
gap> splitting:= [ 1, 2, 3, 4, 5, 6, 7, 8, 8, 9, 10, 11 ];;
gap> split:= CharTableSplitClasses( sub, splitting );;
gap> PrintCharTable( split );
rec( identifier := "Split(Rest(A5.2xS3,[ 1, 3, 4, 6, 7, 9, 10, 12, 14,\
17, 20 ]),[ 1, 2, 3, 4, 5, 6, 7, 8, 8, 9, 10, 11 ])", size :=
360, order :=
360, name := "Split(Rest(A5.2xS3,[ 1, 3, 4, 6, 7, 9, 10, 12, 14, 17, 2\
0 ]),[ 1, 2, 3, 4, 5, 6, 7, 8, 8, 9, 10, 11 ])", centralizers :=
[ 360, 180, 24, 12, 18, 9, 15, 15, 15, 12, 4, 6 ], classes :=
[ 1, 2, 15, 30, 20, 40, 24, 24, 24, 30, 90, 60 ], orders :=
[ 1, 3, 2, 6, 3, 3, 5, 15, 15, 2, 4, 6 ], powermap :=
[ , [ 1, 2, 1, 2, 5, 6, 7, [ 8, 9 ], [ 8, 9 ], 1, 3, 5 ],
[ 1, 1, 3, 3, 1, 1, 7, 7, 7, 10, 11, 10 ],,
[ 1, 2, 3, 4, 5, 6, 1, 2, 2, 10, 11, 12 ] ], irreducibles :=
[ [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ],
[ 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1 ],
[ 2, -1, 2, -1, 2, -1, 2, -1, -1, 0, 0, 0 ],
[ 6, 6, -2, -2, 0, 0, 1, 1, 1, 0, 0, 0 ],
[ 4, 4, 0, 0, 1, 1, -1, -1, -1, 2, 0, -1 ],
[ 4, 4, 0, 0, 1, 1, -1, -1, -1, -2, 0, 1 ],
[ 8, -4, 0, 0, 2, -1, -2, 1, 1, 0, 0, 0 ],
[ 5, 5, 1, 1, -1, -1, 0, 0, 0, 1, -1, 1 ],
[ 5, 5, 1, 1, -1, -1, 0, 0, 0, -1, 1, -1 ],
[ 10, -5, 2, -1, -2, 1, 0, 0, 0, 0, 0, 0 ] ], fusions := [ rec(
name := "Rest(A5.2xS3,[ 1, 3, 4, 6, 7, 9, 10, 12, 14, 17, 20 ])"
,
map := [ 1, 2, 3, 4, 5, 6, 7, 8, 8, 9, 10, 11 ] )
], operations := CharTableOps )
gap> Restricted( sub, split, red );
[ [ 12, -6, -4, 2, 0, 0, 2, -1, -1, 0, 0, 0 ] ]
To complete the table means to find the missing two irreducibles and to complete the powermaps. For this, there are different possibilities. First, one can try to embed G in A_8.
gap> a8:= CharTable( "A8" );;
gap> fus:= SubgroupFusions( split, a8 );
[ [ 1, 4, 3, 9, 4, 5, 8, 13, 14, 3, 7, 9 ],
[ 1, 4, 3, 9, 4, 5, 8, 14, 13, 3, 7, 9 ] ]
gap> fus:= RepresentativesFusions( split, fus, a8 );
#I RepresentativesFusions: no subtable automorphisms stored
[ [ 1, 4, 3, 9, 4, 5, 8, 13, 14, 3, 7, 9 ] ]
gap> StoreFusion( split, a8, fus[1] );
The subgroup fusion is unique up to table automorphisms. Now we restrict the irreducibles of A_8 to G and reduce.
gap> rest:= Restricted( a8, split, a8.irreducibles );;
gap> red:= Reduced( split, split.irreducibles, rest );
rec(
remainders := [ ],
irreducibles :=
[ [ 6, -3, -2, 1, 0, 0, 1, -E(15)-E(15)^2-E(15)^4-E(15)^8,
-E(15)^7-E(15)^11-E(15)^13-E(15)^14, 0, 0, 0 ],
[ 6, -3, -2, 1, 0, 0, 1, -E(15)^7-E(15)^11-E(15)^13-E(15)^14,
-E(15)-E(15)^2-E(15)^4-E(15)^8, 0, 0, 0 ] ] )
gap> Append( split.irreducibles, red.irreducibles );
The list of irreducibles is now complete, but the powermaps are not yet adjusted. To complete the 2nd powermap, we transfer that of A_8 to G using the subgroup fusion.
gap> split.powermap;
[ , [ 1, 2, 1, 2, 5, 6, 7, [ 8, 9 ], [ 8, 9 ], 1, 3, 5 ],
[ 1, 1, 3, 3, 1, 1, 7, 7, 7, 10, 11, 10 ],,
[ 1, 2, 3, 4, 5, 6, 1, 2, 2, 10, 11, 12 ] ]
gap> TransferDiagram( split.powermap[2], fus[1], a8.powermap[2] );;
And this is the complete table.
gap> split.identifier:= "(A5x3):2";;
gap> DisplayCharTable( split );
Split(Rest(A5.2xS3,[ 1, 3, 4, 6, 7, 9, 10, 12, 14, 17, 20 ]),[ 1, 2, 3\
, 4, 5, 6, 7, 8, 8, 9, 10, 11 ])
2 3 2 3 2 1 . . . . 2 2 1
3 2 2 1 1 2 2 1 1 1 1 . 1
5 1 1 . . . . 1 1 1 . . .
1a 3a 2a 6a 3b 3c 5a 15a 15b 2b 4a 6b
2P 1a 3a 1a 3a 3b 3c 5a 15a 15b 1a 2a 3b
3P 1a 1a 2a 2a 1a 1a 5a 5a 5a 2b 4a 2b
5P 1a 3a 2a 6a 3b 3c 1a 3a 3a 2b 4a 6b
X.1 1 1 1 1 1 1 1 1 1 1 1 1
X.2 1 1 1 1 1 1 1 1 1 -1 -1 -1
X.3 2 -1 2 -1 2 -1 2 -1 -1 . . .
X.4 6 6 -2 -2 . . 1 1 1 . . .
X.5 4 4 . . 1 1 -1 -1 -1 2 . -1
X.6 4 4 . . 1 1 -1 -1 -1 -2 . 1
X.7 8 -4 . . 2 -1 -2 1 1 . . .
X.8 5 5 1 1 -1 -1 . . . 1 -1 1
X.9 5 5 1 1 -1 -1 . . . -1 1 -1
X.10 10 -5 2 -1 -2 1 . . . . . .
X.11 6 -3 -2 1 . . 1 A /A . . .
X.12 6 -3 -2 1 . . 1 /A A . . .
A = -E(15)-E(15)^2-E(15)^4-E(15)^8
= (-1-ER(-15))/2 = -1-b15
There are many ways around the block, so two further methods to complete
the table split shall be demonstrated; but we will not go into details.
Without use of GAP one could work as follows:
The irrationalities --and there must be irrational entries in the character table of G, since the outer 2 can conjugate at most two of the four Galois conjugate classes of elements of order 15-- could also have been found from the structure of G and the restriction of the irreducible S_5times S_3 character of degree 12.
On the classes that did not split the values of this character must just
be divided by 2. Let x be one of the irrationalities. The second
orthogonality relation tells us that xcdotoverline{x} = 4 (at class
15a) and x + xast = -1 (at classes 1a and 15a); here xast
denotes the nontrivial Galois conjugate of x. This has no solution for
x = overline{x}, otherwise it leads to the quadratic equation
x^2+x+4 = 0 with solutions b15 = frac{1}{2}(-1+sqrt{-15}) and
-1-b15.
The third possibility to complete the table is to embed A_5times 3:
gap> split.irreducibles := split.irreducibles{ [ 1 .. 10 ] };;
gap> SubgroupFusions( a5xc3, split );
[ [ 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, [ 8, 9 ], [ 8, 9 ], 7, [ 8, 9 ],
[ 8, 9 ] ] ]
The images of the four classes of element order 15 are not determined, the returned list parametrizes the 2^4 possibilities.
gap> fus:= ContainedMaps( last[1] );;
gap> Length( fus );
16
gap> fus[1];
[ 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 7, 8, 8 ]
Most of these 16 possibilities are excluded using scalar products of
induced characters. We take a suitable character chi of a5xc3 and
compute the norm of the induced character with respect to each possible
map.
gap> chi:= a5xc3.irreducibles[5];
[ 3, 3*E(3), 3*E(3)^2, -1, -E(3), -E(3)^2, 0, 0, 0, -E(5)-E(5)^4,
-E(15)^2-E(15)^8, -E(15)^7-E(15)^13, -E(5)^2-E(5)^3,
-E(15)^11-E(15)^14, -E(15)-E(15)^4 ]
gap> List( fus, x -> List( Induced( a5xc3, split, [ chi ], x ),
> y -> ScalarProduct( split, y, y ) )[1] );
[ 8/15, -2/3*E(5)-11/15*E(5)^2-11/15*E(5)^3-2/3*E(5)^4,
-2/3*E(5)-11/15*E(5)^2-11/15*E(5)^3-2/3*E(5)^4, 2/3,
-11/15*E(5)-2/3*E(5)^2-2/3*E(5)^3-11/15*E(5)^4, 3/5, 1,
-11/15*E(5)-2/3*E(5)^2-2/3*E(5)^3-11/15*E(5)^4,
-11/15*E(5)-2/3*E(5)^2-2/3*E(5)^3-11/15*E(5)^4, 1, 3/5,
-11/15*E(5)-2/3*E(5)^2-2/3*E(5)^3-11/15*E(5)^4, 2/3,
-2/3*E(5)-11/15*E(5)^2-11/15*E(5)^3-2/3*E(5)^4,
-2/3*E(5)-11/15*E(5)^2-11/15*E(5)^3-2/3*E(5)^4, 8/15 ]
gap> Filtered( [ 1 .. Length( fus ) ], x -> IsInt( last[x] ) );
[ 7, 10 ]
So only fusions 7 and 10 may be possible. They are equivalent (with respect to table automorphisms), and the list of induced characters contains the missing irreducibles of G:
gap> Sublist( fus, last );
[ [ 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 9, 7, 9, 8 ],
[ 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 9, 8, 7, 8, 9 ] ]
gap> ind:= Induced( a5xc3, split, a5xc3.irreducibles, last[1] );;
gap> Reduced( split, split.irreducibles, ind );
rec(
remainders := [ ],
irreducibles :=
[ [ 6, -3, -2, 1, 0, 0, 1, -E(15)-E(15)^2-E(15)^4-E(15)^8,
-E(15)^7-E(15)^11-E(15)^13-E(15)^14, 0, 0, 0 ],
[ 6, -3, -2, 1, 0, 0, 1, -E(15)^7-E(15)^11-E(15)^13-E(15)^14,
-E(15)-E(15)^2-E(15)^4-E(15)^8, 0, 0, 0 ] ] )
The following example is thought mainly for experts. It shall demonstrate how one can work together with GAP and the ATLAS~CCN85, so better leave out the rest of this section if you are not familiar with the ATLAS.
setlengthunitlength0.1cm We shall construct the character table of the group G = A_6.2^2 cong Aut( A_6 ) from the tables of the normal subgroups A_6.2_1 cong S_6, A_6.2_2 cong PGL(2,9) and A_6.2_3 cong M_{10}.
We regard G as a downward extension of the Klein fourgroup 2^2 with A_6. The set of classes of all preimages of cyclic subgroups of 2^2 covers the classes of G, but it may happen that some representatives are conjugate in G, i.e., the classes fuse.
The ATLAS denotes the character tables of G, G.2_1, G.2_2 and G.2_3 as follows: put(5,-40)beginpicture(50,45) put(25,5)circle1 put(25,20)makebox(0,0)A_6 put(25,30)makebox(0,0)A_6.2_3 put(15,30)makebox(0,0)A_6.2_1 put(35,30)makebox(0,0)A_6.2_2 put(25,40)makebox(0,0)G put(25,5)line(0,1)12 put(23,22)line(-1,1)6 put(27,22)line(1,1)6 put(25,22)line(0,1)6 put(23,38)line(-1,-1)6 put(27,38)line(1,-1)6 put(25,38)line(0,-1)6
vbox
; @ @ @ @ @ @ @ ; ; @ @ @ @ @mboxttchar13360 8 9 9 4 5 5 24 24 4 3 3 p power A A A A A A A A A AB BC p
part A A A A A A A A A AB BC ind 1A 2A 3A 3B 4A 5A B* fus ind 2B 2C 4B 6A 6Bchi_1
+ 1 1 1 1 1 1 1 : ++ 1 1 1 1 1chi_2
+ 5 1 2 -1 -1 0 0 : ++ 3 -1 1 0 -1chi_3
+ 5 1 -1 2 -1 0 0 : ++ -1 3 1 -1 0chi_4
+ 8 0 -1 -1 0 -b5 * . + 0 0 0 0 0chi_5
+ 8 0 -1 -1 0 * -b5 .chi_6
+ 9 1 0 0 1 -1 -1 : ++ 3 3 -1 0 0chi_7
+ 10 -2 1 1 0 0 0 : ++ 2 -2 0 -1 1
vbox put(6.5,-52)line(0,1)8.5 put(56.25,-69)line(0,1)9 put(56.25,-52)line(0,1)8.5
; ; @ @ @ @ @ ; ; @ @ @chi_110 4 4 5 5 2 4 4 A A A BD AD A A A A A A AD BD A A A fus ind 2D 8A B* 10A B* fus ind 4C 8C D**
: ++ 1 1 1 1 1 : ++ 1 1 1
chi_2. + 0 0 0 0 0 . + 0 0 0
chi_3. .
chi_4: ++ 2 0 0 b5 * . + 0 0 0
chi_5: ++ 2 0 0 * b5 .
chi_6: ++ -1 1 1 -1 -1 : ++ 1 -1 -1
chi_7: ++ 0 r2 -r2 0 0 : oo 0 i2 -i2
First we construct a table whose classes are those of the three subgroups. Note that the exponent of A_6 is 60, so the representative orders could become at most 60 times the value in 2^2.
gap> s1:= CharTable( "A6.2_1" );;
gap> s2:= CharTable( "A6.2_2" );;
gap> s3:= CharTable( "A6.2_3" );;
gap> c2:= CharTable( "Cyclic", 2 );;
gap> v4:= CharTableDirectProduct( c2, c2 );;
#I CharTableDirectProduct: existing subgroup fusion on <tbl2> replaced
#I by actual one
gap> for tbl in [ s1, s2, s3 ] do
> Print( tbl.irreducibles[2], "\n" );
> od;
[ 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> split:= CharTableSplitClasses( v4,
> [1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4], 60 );;
gap> PrintCharTable( split );
rec( identifier := "Split(C2xC2,[ 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, \
3, 3, 3, 4, 4, 4 ])", size := 4, order :=
4, name := "Split(C2xC2,[ 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3,\
4, 4, 4 ])", centralizers := [ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,
4, 4, 4, 4, 4, 4 ], classes := [ 1/5, 1/5, 1/5, 1/5, 1/5, 1/5, 1/5,
1/5, 1/5, 1/5, 1/5, 1/5, 1/5, 1/5, 1/5, 1/3, 1/3, 1/3 ], orders :=
[ 1, [ 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ],
[ 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ],
[ 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ],
[ 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ],
[ 2, 4, 6, 8, 10, 12, 20, 24, 30, 40, 60, 120 ],
[ 2, 4, 6, 8, 10, 12, 20, 24, 30, 40, 60, 120 ],
[ 2, 4, 6, 8, 10, 12, 20, 24, 30, 40, 60, 120 ],
[ 2, 4, 6, 8, 10, 12, 20, 24, 30, 40, 60, 120 ],
[ 2, 4, 6, 8, 10, 12, 20, 24, 30, 40, 60, 120 ],
[ 2, 4, 6, 8, 10, 12, 20, 24, 30, 40, 60, 120 ],
[ 2, 4, 6, 8, 10, 12, 20, 24, 30, 40, 60, 120 ],
[ 2, 4, 6, 8, 10, 12, 20, 24, 30, 40, 60, 120 ],
[ 2, 4, 6, 8, 10, 12, 20, 24, 30, 40, 60, 120 ],
[ 2, 4, 6, 8, 10, 12, 20, 24, 30, 40, 60, 120 ],
[ 2, 4, 6, 8, 10, 12, 20, 24, 30, 40, 60, 120 ],
[ 2, 4, 6, 8, 10, 12, 20, 24, 30, 40, 60, 120 ],
[ 2, 4, 6, 8, 10, 12, 20, 24, 30, 40, 60, 120 ] ], powermap :=
[ , [ 1, [ 1, 2, 3, 4, 5 ], [ 1, 2, 3, 4, 5 ], [ 1, 2, 3, 4, 5 ],
[ 1, 2, 3, 4, 5 ], [ 1, 2, 3, 4, 5 ], [ 1, 2, 3, 4, 5 ],
[ 1, 2, 3, 4, 5 ], [ 1, 2, 3, 4, 5 ], [ 1, 2, 3, 4, 5 ],
[ 1, 2, 3, 4, 5 ], [ 1, 2, 3, 4, 5 ], [ 1, 2, 3, 4, 5 ],
[ 1, 2, 3, 4, 5 ], [ 1, 2, 3, 4, 5 ], [ 1, 2, 3, 4, 5 ],
[ 1, 2, 3, 4, 5 ], [ 1, 2, 3, 4, 5 ] ] ], irreducibles :=
[ [ 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, -1 ],
[ 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1 ]
], fusions := [ rec(
name := [ 'C', '2', 'x', 'C', '2' ],
map := [ 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4 ]
) ], operations := CharTableOps )
Now we embed the subgroups and adjust the classlengths, order, centralizers, powermaps and thus the representative orders.
gap> StoreFusion( s1, split, [1,2,3,3,4,5,6,7,8,9,10]);
gap> StoreFusion( s2, split, [1,2,3,4,5,5,11,12,13,14,15]);
gap> StoreFusion( s3, split, [1,2,3,4,5,16,17,18]);
gap> for tbl in [ s1, s2, s3 ] do
> fus:= GetFusionMap( tbl, split );
> for class in Difference( [ 1 .. Length( tbl.classes ) ],
> KernelChar(tbl.irreducibles[2]) ) do
> split.classes[ fus[ class ] ]:= tbl.classes[ class ];
> od;
> od;
gap> for class in [ 1 .. 5 ] do
> split.classes[ class ]:= s3.classes[ class ];
> od;
gap> split.classes;
[ 1, 45, 80, 90, 144, 15, 15, 90, 120, 120, 36, 90, 90, 72, 72, 180,
90, 90 ]
gap> split.size:= Sum( last );
1440
gap> split.order:= last;
gap> split.centralizers:= List( split.classes, x -> split.order / x );
[ 1440, 32, 18, 16, 10, 96, 96, 16, 12, 12, 40, 16, 16, 20, 20, 8,
16, 16 ]
gap> split.powermap[3]:= InitPowermap( split, 3 );;
gap> split.powermap[5]:= InitPowermap( split, 5 );;
gap> for tbl in [ s1, s2, s3 ] do
> fus:= GetFusionMap( tbl, split );
> for p in [ 2, 3, 5 ] do
> TransferDiagram( tbl.powermap[p], fus, split.powermap[p] );
> od;
> od;
gap> split.powermap;
[ , [ 1, 1, 3, 2, 5, 1, 1, 2, 3, 3, 1, 4, 4, 5, 5, 2, 4, 4 ],
[ 1, 2, 1, 4, 5, 6, 7, 8, 6, 7, 11, 13, 12, 15, 14, 16, 17, 18 ],,
[ 1, 2, 3, 4, 1, 6, 7, 8, 9, 10, 11, 13, 12, 11, 11, 16, 18, 17 ] ]
gap> split.orders:= ElementOrdersPowermap( split.powermap );
[ 1, 2, 3, 4, 5, 2, 2, 4, 6, 6, 2, 8, 8, 10, 10, 4, 8, 8 ]
In order to decide which classes fuse in G, we look at the norms of suitable induced characters, first the + extension of chi_2 to A_6.2_1.
gap> ind:= Induced( s1, split, [ s1.irreducibles[3] ] )[1];
[ 10, 2, 1, -2, 0, 6, -2, 2, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0 ]
gap> ScalarProduct( split, ind, ind );
3/2
The inertia group of this character is A_6.2_1, thus the norm of the
induced character must be 1. If the classes 2B and 2C fuse, the
contribution of these classes is changed from 15cdot 6^2+15cdot(-2)^2
to 30 cdot 2^2, the difference is 480. But we have to subtract 720
which is half the group order, so also 6A and 6B fuse. This is not
surprising, since it reflects the action of the famous outer automorphism
of S_6. Next we examine the + extension of chi_4 to A_6.2_2.
gap> ind:= Induced( s2, split, [ s2.irreducibles[4] ] )[1];
[ 16, 0, -2, 0, 1, 0, 0, 0, 0, 0, 4, 0, 0, 2*E(5)+2*E(5)^4,
2*E(5)^2+2*E(5)^3, 0, 0, 0 ]
gap> ScalarProduct( split, ind, ind );
3/2
Again, the norm must be 1, 10A and 10B fuse.
gap> collaps:= CharTableCollapsedClasses( split,
> [1,2,3,4,5,6,6,7,8,8,9,10,11,12,12,13,14,15] );;
gap> PrintCharTable( collaps );
rec( identifier := "Collapsed(Split(C2xC2,[ 1, 1, 1, 1, 1, 2, 2, 2, 2,\
2, 3, 3, 3, 3, 3, 4, 4, 4 ]),[ 1, 2, 3, 4, 5, 6, 6, 7, 8, 8, 9, 10, 1\
1, 12, 12, 13, 14, 15 ])", size := 1440, order :=
1440, name := "Collapsed(Split(C2xC2,[ 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3\
, 3, 3, 3, 3, 4, 4, 4 ]),[ 1, 2, 3, 4, 5, 6, 6, 7, 8, 8, 9, 10, 11, 12\
, 12, 13, 14, 15 ])", centralizers := [ 1440, 32, 18, 16, 10, 48, 16,
6, 40, 16, 16, 10, 8, 16, 16 ], orders :=
[ 1, 2, 3, 4, 5, 2, 4, 6, 2, 8, 8, 10, 4, 8, 8 ], powermap :=
[ , [ 1, 1, 3, 2, 5, 1, 2, 3, 1, 4, 4, 5, 2, 4, 4 ],
[ 1, 2, 1, 4, 5, 6, 7, 6, 9, 11, 10, 12, 13, 14, 15 ],,
[ 1, 2, 3, 4, 1, 6, 7, 8, 9, 11, 10, 9, 13, 15, 14 ]
], fusionsource :=
[ "Split(C2xC2,[ 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4 \
])" ], irreducibles :=
[ [ 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, -1, -1, -1, -1, 1, 1, 1 ]
], classes := [ 1, 45, 80, 90, 144, 30, 90, 240, 36, 90, 90, 144,
180, 90, 90 ], operations := CharTableOps )
gap> split.fusions;
[ rec(
name := [ 'C', '2', 'x', 'C', '2' ],
map := [ 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4 ]
), rec(
name :=
"Collapsed(Split(C2xC2,[ 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3,\
3, 3, 4, 4, 4 ]),[ 1, 2, 3, 4, 5, 6, 6, 7, 8, 8, 9, 10, 11, 12, 12, 1\
3, 14, 15 ])",
map := [ 1, 2, 3, 4, 5, 6, 6, 7, 8, 8, 9, 10, 11, 12, 12, 13,
14, 15 ] ) ]
gap> for tbl in [ s1, s2, s3 ] do
> StoreFusion( tbl, collaps,
> CompositionMaps( GetFusionMap( split, collaps ),
> GetFusionMap( tbl, split ) ) );
> od;
gap> ind:= Induced( s1, collaps, [ s1.irreducibles[10] ] )[1];
[ 20, -4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
gap> ScalarProduct( collaps, ind, ind );
1
This character must be equal to any induced character of an irreducible
character of degree 10 of A_6.2_2 and A_6.2_3. That means, 8A
fuses with 8B, and 8C with 8D.
gap> a6v4:= CharTableCollapsedClasses( collaps,
> [1,2,3,4,5,6,7,8,9,10,10,11,12,13,13] );;
gap> PrintCharTable( a6v4 );
rec( identifier := "Collapsed(Collapsed(Split(C2xC2,[ 1, 1, 1, 1, 1, 2\
, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4 ]),[ 1, 2, 3, 4, 5, 6, 6, 7, 8, 8\
, 9, 10, 11, 12, 12, 13, 14, 15 ]),[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10\
, 11, 12, 13, 13 ])", size := 1440, order :=
1440, name := "Collapsed(Collapsed(Split(C2xC2,[ 1, 1, 1, 1, 1, 2, 2, \
2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4 ]),[ 1, 2, 3, 4, 5, 6, 6, 7, 8, 8, 9, \
10, 11, 12, 12, 13, 14, 15 ]),[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 11,\
12, 13, 13 ])", centralizers := [ 1440, 32, 18, 16, 10, 48, 16, 6,
40, 8, 10, 8, 8 ], orders := [ 1, 2, 3, 4, 5, 2, 4, 6, 2, 8, 10, 4,
8 ], powermap := [ , [ 1, 1, 3, 2, 5, 1, 2, 3, 1, 4, 5, 2, 4 ],
[ 1, 2, 1, 4, 5, 6, 7, 6, 9, 10, 11, 12, 13 ],,
[ 1, 2, 3, 4, 1, 6, 7, 8, 9, 10, 9, 12, 13 ] ], fusionsource :=
[ "Collapsed(Split(C2xC2,[ 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3\
, 4, 4, 4 ]),[ 1, 2, 3, 4, 5, 6, 6, 7, 8, 8, 9, 10, 11, 12, 12, 13, 14\
, 15 ])" ], irreducibles :=
[ [ 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 ] ], classes :=
[ 1, 45, 80, 90, 144, 30, 90, 240, 36, 180, 144, 180, 180
], operations := CharTableOps )
gap> for tbl in [ s1, s2, s3 ] do
> StoreFusion( tbl, a6v4,
> CompositionMaps( GetFusionMap( collaps, a6v4 ),
> GetFusionMap( tbl, collaps ) ) );
> od;
Now the classes of G are known, the only remaining work is to compute the irreducibles.
gap> a6v4.irreducibles;
[ [ 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> for tbl in [ s1, s2, s3 ] do
> ind:= Set( Induced( tbl, a6v4, tbl.irreducibles ) );
> Append( a6v4.irreducibles,
> Filtered( ind, x -> ScalarProduct( a6v4,x,x ) = 1 ) );
> od;
gap> a6v4.irreducibles:= Set( a6v4.irreducibles );
[ [ 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 ],
[ 10, 2, 1, -2, 0, -2, -2, 1, 0, 0, 0, 0, 0 ],
[ 10, 2, 1, -2, 0, 2, 2, -1, 0, 0, 0, 0, 0 ],
[ 16, 0, -2, 0, 1, 0, 0, 0, -4, 0, 1, 0, 0 ],
[ 16, 0, -2, 0, 1, 0, 0, 0, 4, 0, -1, 0, 0 ],
[ 20, -4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ]
gap> sym:= Symmetrizations( a6v4, [ a6v4.irreducibles[5] ], 2 );
[ [ 45, -3, 0, 1, 0, -3, 1, 0, -5, 1, 0, -1, 1 ],
[ 55, 7, 1, 3, 0, 7, 3, 1, 5, -1, 0, 1, -1 ] ]
gap> Reduced( a6v4, a6v4.irreducibles, sym );
rec(
remainders := [ [ 27, 3, 0, 3, -3, 3, -1, 0, 1, -1, 1, 1, -1 ] ],
irreducibles := [ [ 9, 1, 0, 1, -1, -3, 1, 0, -1, 1, -1, -1, 1 ] ] )
gap> Append( a6v4.irreducibles,
> Tensored( last.irreducibles,
> Sublist( a6v4.irreducibles, [ 1 .. 4 ] ) ) );
gap> SortCharactersCharTable( a6v4,
> (1,4)(2,3)(5,6)(7,8)(9,13,10,11,12) );;
gap> a6v4.identifier:= "A6.2^2";;
gap> DisplayCharTable( a6v4 );
Collapsed(Collapsed(Split(C2xC2,[ 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, \
3, 3, 3, 4, 4, 4 ]),[ 1, 2, 3, 4, 5, 6, 6, 7, 8, 8, 9, 10, 11, 12, 12,\
13, 14, 15 ]),[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 11, 12, 13, 13 ])
2 5 5 1 4 1 4 4 1 3 3 1 3 3
3 2 . 2 . . 1 . 1 . . . . .
5 1 . . . 1 . . . 1 . 1 . .
1a 2a 3a 4a 5a 2b 4b 6a 2c 8a 10a 4c 8b
2P 1a 1a 3a 2a 5a 1a 2a 3a 1a 4a 5a 2a 4a
3P 1a 2a 1a 4a 5a 2b 4b 2b 2c 8a 10a 4c 8b
5P 1a 2a 3a 4a 1a 2b 4b 6a 2c 8a 2c 4c 8b
X.1 1 1 1 1 1 1 1 1 1 1 1 1 1
X.2 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1
X.3 1 1 1 1 1 -1 -1 -1 1 1 1 -1 -1
X.4 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1
X.5 10 2 1 -2 . 2 2 -1 . . . . .
X.6 10 2 1 -2 . -2 -2 1 . . . . .
X.7 16 . -2 . 1 . . . 4 . -1 . .
X.8 16 . -2 . 1 . . . -4 . 1 . .
X.9 9 1 . 1 -1 -3 1 . 1 -1 1 1 -1
X.10 9 1 . 1 -1 -3 1 . -1 1 -1 -1 1
X.11 9 1 . 1 -1 3 -1 . 1 -1 1 -1 1
X.12 9 1 . 1 -1 3 -1 . -1 1 -1 1 -1
X.13 20 -4 2 . . . . . . . . . .
GAP 3.4.4