User:Tomruen/Coxeter foldings

Coxeter group	Coxeter diagram	Degrees	Coxeter planes
A2		2, 3	A1, A2
B2		2, 4	A1, B2
H2		2, 5	A1, H2
A3		2, 3, 4	A1, A2, A3
B3		2, 4, 6	A1, B2, A2=B3
H3		2, 6, 10	A1, A2, H2=H3
A4		2, 3, 4, 5	A1, A2, A3, A4
B4		2, 4, 6, 8	A1, A3, B2, A2=B3, B4
D4		2, 4, 6	A1, A3, A2=D4
F4		2, 6, 8, 12	A1, A3=B2, A2=B3, F4
H4		2, 12, 20, 30	A1, A2, A3, H2=H3, H4
A5		2, 3, 4, 5, 6	A1, A2, A3, A4, A5
B5		2, 4, 6, 8, 10	A1, A3=B2, A2=B3, B4, A4=B5
D5		2, 4, 6, 8; 5	A1, A3, A2=D4, D5; A4
A6		2, 3, 4, 5, 6, 7	A1, A2, A3, A4, A5, A6
B6		2, 4, 6, 8, 10, 12	A1, A3=B2, A2=B3, B4, A4=B5, B6
D6		2, 4, 6, 8, 10
E6		2, 5, 6, 8, 9, 12	A1, A4, A2=D4=A5, A3=D5, ?, E6
E7		2, 6, 8, 10, 12, 14, 18
E8		2, 8, 12, 14, 18, 20, 24, 30

Let me try using Coxeter–Dynkin_diagram#Geometric_foldings to express Coxeter planes as Coxeter numbers and all degrees of fundamental invariants. Foldings are shown by marking node with colors, re and blue, which map to node 1 or 2 in the rank 2 folded group.

Example: A₃,

Folding		Degree	Coxeter Plane
		4	A3
		3	A2
		2	A1

Example: B₃,

Folding		Degree	Coxeter Plane
		6	B3
		3×2	A2
		4	B2
		2	A1

Example: H₃,

Folding		Degree	Coxeter Plane
		10	H3
		5×2	H2
		3×2	A2
		2	A1

Example: A₄,

Folding			Degree	Coxeter Plane
			5	A4
			4	A3
			3	A2
			2	A1

Example: B₄,

Folding			Degree	Coxeter Plane
			8	B4
			6	B3
			3×2	A2
			4	A3
			4	B2
			2	A1

Example: D₄,

Folding			Degree	Coxeter Plane
			6	D4=B3
			3×2	A2
	=		4	D3=A3
			4	B2
			2	A1

Example: F₄,

Folding			Degree	Coxeter Plane
			12	F4
			4×2	A3
			4×2	B2
			6	B3
			3×2	A2
			2	A1

Example: H₄,

Folding			Degree	Coxeter Plane
			30	H4
			20
			12	F4
			10	H3
			5×2	H2
			3×2	A2
			4	A3
			2	A1

Example: A₅,

Folding			Degree	Coxeter Plane
			6	A5
			5	A4
			4	A3
			3	A2
			2	A1

Example: B₅,

Folding			Degree	Coxeter Plane
			10	B5
			5×2	A4
			8	B4
			6	B3
			3×2	A2
			4	A3
			4	B2
			2	A1

Example: D₅,

Folding			Degree	Coxeter Plane
			8	D5=B4
	=		6	D4=B3
			3×2	A2
			5	A4
	=		4	D3=A3
			2	A1

Example: E₆,

Folding			Degree	Coxeter Plane
			12	E6 = F4
			9
	=		8	D5 = B4
			6	A5
	=		6	D4 = B3
			3×2	A2
			5	A4
			4	A3
			2	A1