Erroneous characters from orthogonality in C2 x SL(2,3) and SL(2,3) x: C2

[aidevnn]

FastGoat/FastGoat/Examples/GroupMatrixFormPart2.cs

Lines 669 to 682 in b3bab8f

	public static void ExampleNonIsotypicDecomposition()
	{
	GlobalStopWatch.Restart();
	var c2sl23 = FG.WordGroup("C2 x SL(2,3)", "a4, b3, c2, ababab, caca-1, cbcb-1, a2ba2b-1");
	var sl23byc2 = FG.WordGroup("SL(2,3) x: C2", "a4, c3, a2b2, abab, acacac, cbc-1b-1");
	foreach (var (g, mtGL, matSubgrs, names) in new[] { c2sl23, sl23byc2 }.Select(sg => MatrixFormOfGroup(sg)))
	{
	FG.DisplayName(mtGL, matSubgrs, names, false, false, true, 20);
	GetCharacter(mtGL, matSubgrs);
	}
	GlobalStopWatch.Show("END");
	Console.Beep();
	}

FastGoat/FastGoat/Examples/GroupMatrixFormPart2.cs

Lines 530 to 531 in b3bab8f

	else
	ct.SolveOrthogonality((2, 6.Range()));

The characters resulting from solving orthogonality are erroneous and lead to an incorrect decomposition of the representation.

[aidevnn]

When limiting the search to two missing characters, incorrect results may occur for example in SL(2,3).

FastGoat/FastGoat/DetailsGroupsUptoOrder32.txt

Lines 4130 to 4147 in 835336c

	Reductible Representation
	ρ = 4 - 4(2) + (3a) + (3b) - (6a) - (6b)
	3 Isotypic components
	ρ = 4/3Ꭓ.4 + 1/3Ꭓ.5 + 1/3Ꭓ.6
	Character Table
	[Class 1 2 3a 3b 4 6a 6b]
	[ Size 1 1 4 4 6 4 4]
	[ ]
	[ Ꭓ.1 1 1 1 1 1 1 1]
	[ Ꭓ.2 1 1 ξ3 ξ3^2 1 ξ3^2 ξ3]
	[ Ꭓ.3 1 1 ξ3^2 ξ3 1 ξ3 ξ3^2]
	[ Ꭓ.4 2 -2 1 1 0 -1 -1]
	[ Ꭓ.5 2 -2 ξ3 ξ3^2 0 -ξ3^2 -ξ3]
	[ Ꭓ.6 2 -2 ξ3^2 ξ3 0 -ξ3 -ξ3^2]
	[ Ꭓ.7 3 3 0 0 -1 0 0]

This is because the equations used are only necessary conditions for determining irreducible characters basis from class functions, as <Xi,Xi> = 1. To ensure accuracy, additional conditions must be applied beforehand or a more robust validation of solutions should be conducted afterward.

[aidevnn]

The symmetric and exterior characters appear to function as a validation method when they can undergo an isotypic decomposition. This is applicable for solutions to the orthogonality equations for missing characters of groups such as SL(2,3), C2 x SL(2,3), and SL(2,3) x: C2.

Further testing is required for additional groups.

[aidevnn]

The previous remark applies solely to SL(2,3) when determining missing characters.

For the groups C2 x SL(2,3) and SL(2,3) x: C2, certain class functions that solve the system and yield integer decompositions for symmetric characters may not be valid. These solutions do not necessarily result in integer decompositions for representations in the GroupMatrixFormPart2.cs file.

Occasionally, fortuitous ordering can yield satisfactory results for the previous cases.