DualCoxeterNumber

The dual Coxeter number of the type.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001154via CartanType✓↗

DualCoxeterNumber(cartanType)The dual Coxeter number of the type.

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 * _x[2] - 1 if _x[1] == "B" else _x[2] + 1 if _x[1] ==
                                                                               "C" else 2 * _x[2] -
                                                                                     2 if _x[1] == "D" else (12 if _x[2] == 6 else 18 if _x[2] == 7 else 30) if _x[1] == "E" else 9 if _x[1] == "F" else 4

Examples

Basic #

the type $D_5$

live ↗

In$\mathrm{DualCoxeterNumber}(\operatorname{CartanType}((\text{D},5)))$Out$8$

the type $E_6$

live ↗

In$\mathrm{DualCoxeterNumber}(\operatorname{CartanType}((\text{E},6)))$Out$12$

Properties #

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

live ↗

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

Known value · FindStat St001154

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

live ↗

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

Known value · FindStat St001154

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

live ↗

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

Known value · FindStat St001154

About

Examples	5
Crosswalk	FindStat