"Source: Text/RepThy/Chtr.text";
"Line: 1567";
"Date: Fri Sep 26 12:10:12 2025";
"Main: Fri Sep 26 14:29:19 2025";
// original file: Text/RepThy/Chtr.text, line: 1567
// Example: H100E8 ()
print "Example: H100E8";
ei := GetEchoInput();
SetEchoInput(true);
A := AlternatingGroup(GrpPerm, 5);
R := CharacterRing(A);
T1 := R ! 1;
T1;
pc := PermutationCharacter(A);
T2 := pc - T1;
InnerProduct(pc, T1), InnerProduct(T2, T2);
T2;
B := Stabilizer(A, 5);
r := RootOfUnity(3, CyclotomicField(3));
S := CharacterRing(B);
lambda := S ! [1, 1, r, r^2 ];
IsLinear(lambda);
assert $1;
T3 := Induction(lambda, A);
InnerProduct(T3, T3);
assert $1 eq 1;
T3;
K := sub<A |  (1,2,3,4,5) >;
Y := CharacterTable(K);
Y;
mu := Induction(Y[2], A);
_, T4 := Decomposition([T1, T2, T3], mu);
InnerProduct(T4, T4);
assert $1 eq 1;
T4;
T5 := GaloisConjugate(T4, 2);
T5;
CharacterTable(A);
Q := [<1,1>, <2,15>, <3,20>, <5, 12>, <5,12>];
R := CharacterRing(Q);
x := R![4,0,1,-1,-1];
x`IsIrreducible := true;
x;
#KnownIrreducibles(R);
pm := [[1],[2,1],[3,3,1],[4,5,5,4,1],[5,4,4,5,1]];
R`PowerMap := pm;
y := Symmetrization(x, [2]);
InnerProduct(y, R!1);
y := y - R!1;
InnerProduct(y, x);
y := y - x;
y;
Norm(y);
y`IsIrreducible := true;
#KnownIrreducibles(R);  
SetEchoInput(ei);
