CharTableSplitClasses( tbl, fusionmap )
CharTableSplitClasses( tbl, fusionmap, exponent )
returns a character table where the classes of the character table tbl are split according to the fusion map fusionmap.
The two forms correspond to the two different situations to split classes:
CharTableSplitClasses( tbl, fusionmap )
If one constructs a normal subgroup (see CharTableNormalSubgroup), the order remains unchanged, powermaps, classlengths and centralizer orders are changed with respect to the fusion, representative orders and irreducibles are simply split. The ``factor fusion'' fusionmap to tbl is stored on the result.
# see example in "CharTableNormalSubgroup"
gap> split:= CharTableSplitClasses(sub,[1,2,3,4,5,6,7,8,8,9,10,11]);;
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> # the table of $(3\times A_5)\!\:\! 2$ (incomplete)
CharTableSplitClasses( tbl, fusionmap, exponent )
To construct a downward extension is somewhat more complicated, since the new order, representative orders, centralizer orders and classlengths are not known at the moment when the classes are split. So the order remains unchanged, centralizer orders will just be split, classlengths are divided by the number of image classes, and the representative orders become parametrized with respect to the exponent exponent of the normal subgroup. Power maps and irreducibles are computed from tbl and fusionmap, and the factor fusion fusionmap to tbl is stored on the result.
gap> a5:= CharTable( "Alternating", 5 );;
gap> CharTableSplitClasses( a5, [ 1, 1, 2, 3, 3, 4, 4, 5, 5 ], 2 );;
gap> PrintCharTable( last );
rec( identifier := "Split(A5,[ 1, 1, 2, 3, 3, 4, 4, 5, 5 ])", size :=
60, order :=
60, name := "Split(A5,[ 1, 1, 2, 3, 3, 4, 4, 5, 5 ])", centralizers :=\
[ 60, 60, 4, 3, 3, 5, 5, 5, 5 ], classes :=
[ 1/2, 1/2, 15, 10, 10, 6, 6, 6, 6 ], orders :=
[ 1, 2, [ 2, 4 ], [ 3, 6 ], [ 3, 6 ], [ 5, 10 ], [ 5, 10 ],
[ 5, 10 ], [ 5, 10 ] ], powermap :=
[ , [ 1, 1, [ 1, 2 ], [ 4, 5 ], [ 4, 5 ], [ 8, 9 ], [ 8, 9 ],
[ 6, 7 ], [ 6, 7 ] ],
[ 1, 2, 3, [ 1, 2 ], [ 1, 2 ], [ 8, 9 ], [ 8, 9 ], [ 6, 7 ],
[ 6, 7 ] ],,
[ 1, 2, 3, [ 4, 5 ], [ 4, 5 ], [ 1, 2 ], [ 1, 2 ], [ 1, 2 ],
[ 1, 2 ] ] ], irreducibles := [ [ 1, 1, 1, 1, 1, 1, 1, 1, 1 ],
[ 4, 4, 0, 1, 1, -1, -1, -1, -1 ], [ 5, 5, 1, -1, -1, 0, 0, 0, 0 ],
[ 3, 3, -1, 0, 0, -E(5)-E(5)^4, -E(5)-E(5)^4, -E(5)^2-E(5)^3,
-E(5)^2-E(5)^3 ],
[ 3, 3, -1, 0, 0, -E(5)^2-E(5)^3, -E(5)^2-E(5)^3, -E(5)-E(5)^4,
-E(5)-E(5)^4 ] ], fusions := [ rec(
name := "A5",
map := [ 1, 1, 2, 3, 3, 4, 4, 5, 5 ] )
], operations := CharTableOps )
Note that powermaps (and in the second case also the representative orders) may become parametrized maps (see Chapter Maps and Parametrized Maps).
The inverse process of splitting is the fusion of classes, see CharTableCollapsedClasses.
GAP 3.4.4