Improving Character Table Computation

[aidevnn]

The current implementation for computing character tables is based solely on the following methods:

• Lifting from the derived subgroup
• Induction from subgroups
• Restriction from the supergroup, as in the case of SL(2,3)

This implementation can be enhanced by incorporating various named characters and resolving orthogonality and an old implementation for resolving orthogonality can be retrieved from git-history.

[aidevnn]

When dealing with the character table, some small improvement can be tested

• the regular character to complete the last line,
• an induction from the stabilizer subgroup corresponding to each class,
• additionally, if the stabilizer subgroup is normal, lifting can also be done.

[aidevnn]

In the case of abelian group, induction from its subgroups is x3 slower than direct computation of linear characters from tensor property.

FastGoat/FastGoat/Examples/AllGroupsUptoOrder32.cs

Lines 42 to 49 in b44c844

	{
	var ctG = FG.CharacterTableEmpty(g);
	ctG.DerivedSubGroupLift();
	ctG.InductionFromSubGroups(subgroups);
	ctG.DisplayCells(tableOnly: true);
	Console.WriteLine();
	}
	}

It is now fixed.

FastGoat/FastGoat/Examples/AllGroupsUptoOrder32.cs

Lines 40 to 53 in 982a0a0

	{
	var ctG = FG.CharacterTableEmpty(g);
	if(g.GroupType == GroupType.AbelianGroup)
	ctG.AbelianTable();
	else
	{
	ctG.DerivedSubGroupLift();
	ctG.InductionFromStabilizers();
	ctG.InductionFromSubGroups(subgroups);
	}
	ctG.DisplayCells(tableOnly: true);
	Console.WriteLine();
	}

[aidevnn]

When linear characters are associated with a new character with unknowns, it is possible to reduce and solve the system for SL(2,3), GL(2,3), SL(2,3) x C2, and SL(2,3) x: C2. However, a method to avoid unnecessary linear characters still needs to be found.

[aidevnn]

Pretty print error needs to be fixed, I^2 = -1.

FastGoat/FastGoat/DetailsGroupsUptoOrder32.txt

Lines 6116 to 6119 in 8b55d92

	[ Ꭓ.11 2 -2 0 0 -2*I 2*I -I - I^2 -I + I^2 I + I^2 I - I^2 0 0 0 0]
	[ Ꭓ.12 2 -2 0 0 -2*I 2*I I + I^2 I - I^2 -I - I^2 -I + I^2 0 0 0 0]
	[ Ꭓ.13 2 -2 0 0 2*I -2*I -I + I^2 -I - I^2 I - I^2 I + I^2 0 0 0 0]
	[ Ꭓ.14 2 -2 0 0 2*I -2*I I - I^2 I + I^2 -I + I^2 -I - I^2 0 0 0 0]