SortCharTable( tbl, kernel )
SortCharTable( tbl, normalseries )
SortCharTable( tbl, facttbl, kernel )
sorts classes and irreducibles of the character table tbl, and
returns a record with components columns and rows, which are the
permutations applied to classes and characters.
The first form sorts the classes at positions contained in the list
kernel to the beginning, and sorts all characters in
tbl.irreducibles such that the first characters are those that
contain kernel in their kernel.
The second form does the same successively for all kernels k_i in the list 'normalseries' = [ k_1, k_2, ldots, k_n ] where k_i must be a sublist of k_{i+1} for 1 leq i leq n-1.
The third form computes the table F of the factor group of tbl
modulo the normal subgroup formed by the classes whose positions are
contained in the list kernel;
F must be permutation equivalent to the table facttbl (in the
sense of TransformingPermutationsCharTables), else false is
returned. The classes of tbl are sorted such that the preimages
of a class of F are consecutive, and that the succession of
preimages is that of facttbl. tbl.irreducibles is sorted as
by SortCharTable( tbl, kernel ).
(Note that the transformation is only unique up to table automorphisms
of F, and this need not be unique up to table automorphisms of tbl.)
All rearrangements of classes and characters are stable, i.e., the relative positions of classes and characters that are not distinguished by any relevant property is not changed.
SortCharTable uses SortClassesCharTable SortClassesCharTable and
SortCharactersCharTable SortCharactersCharTable.
gap> t:= CharTable("Symmetric",4);;
gap> Set( List( t.irreducibles, KernelChar ) );
[ [ 1 ], [ 1, 2, 3, 4, 5 ], [ 1, 3 ], [ 1, 3, 4 ] ]
gap> SortCharTable( t, Permuted( last, (2,4,3) ) );
rec(
columns := (2,4,3),
rows := (1,2,4,5) )
gap> DisplayCharTable( t );
S4
2 3 3 . 2 2
3 1 . 1 . .
1a 2a 3a 2b 4a
2P 1a 1a 3a 1a 2a
3P 1a 2a 1a 2b 4a
X.1 1 1 1 1 1
X.2 1 1 1 -1 -1
X.3 2 2 -1 . .
X.4 3 -1 . -1 1
X.5 3 -1 . 1 -1
GAP 3.4.4