CartanMatrixDeterminant

The determinant of the Cartan matrix, the order of the center of the simply connected group.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000821via CartanType✓↗

CartanMatrixDeterminant(cartanType)The determinant of the Cartan matrix, the order of the center of the simply connected group.

Typed signatures

Type	Package	Applies
(cartan_type) -> number	combinatorics	always

Domain: Cartan type statistics

Details

• Defined over CartanType as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a CartanType and nothing else — it reads values against their positions, or walks the orbits, so it needs the bijection. Applying it to a bare list is a type error, not a wrong answer.

Definition

Epsil, as the implementation runs it.

on CartanType, over _x

_x[2] + 1 if _x[1] == "A" else 2 if _x[1] == "B" else 2 if _x[1] == "C" else 4 if _x[1] == "D" else (3 if _x[2] == 6 else 2 if _x[2] == 7 else 1) if _x[1] == "E" else 1 if _x[1] == "F" else 1

Examples

Basic #

the type $D_5$

live ↗

In$\mathrm{CartanMatrixDeterminant}(\operatorname{CartanType}((\text{D},5)))$Out$4$

the type $E_6$

live ↗

In$\mathrm{CartanMatrixDeterminant}(\operatorname{CartanType}((\text{E},6)))$Out$3$

Properties #

FindStat's value for ['A',1]

live ↗

In$\mathrm{CombinatorialStat}(\operatorname{CartanType}(("A",1)), \text{CartanMatrixDeterminant})$Out$2$

Known value · FindStat St000821

FindStat's value for ['A',2]

live ↗

In$\mathrm{CombinatorialStat}(\operatorname{CartanType}(("A",2)), \text{CartanMatrixDeterminant})$Out$3$

Known value · FindStat St000821

FindStat's value for ['A',3]

live ↗

In$\mathrm{CombinatorialStat}(\operatorname{CartanType}(("A",3)), \text{CartanMatrixDeterminant})$Out$4$

Known value · FindStat St000821

About

Examples	5
Crosswalk	FindStat