ReflectionOrbitCount

The conjugacy classes of reflections in the Weyl group: 2 where the simple roots have two lengths (B, C, F, G), else 1.

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ReflectionOrbitCount(cartanType)The conjugacy classes of reflections in the Weyl group: 2 where the simple roots have two lengths (B, C, F, G), else 1.

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

1 if _x[1] == "A" else 2 if _x[1] == "B" else 2 if _x[1] == "C" else 1 if _x[1] ==
                                                                          "D" else 1 if _x[1] == "E" else 2 if _x[1] == "F" else 2

Examples

Basic #

the type $D_5$

live ↗

In$\mathrm{ReflectionOrbitCount}(\operatorname{CartanType}((\text{D},5)))$Out$1$

the type $E_6$

live ↗

In$\mathrm{ReflectionOrbitCount}(\operatorname{CartanType}((\text{E},6)))$Out$1$

Properties #

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

live ↗

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

Known value · FindStat St000854

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

live ↗

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

Known value · FindStat St000854

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

live ↗

In$\mathrm{CombinatorialStat}(\operatorname{CartanType}(("A",3)), \text{ReflectionOrbitCount})$Out$1$

Known value · FindStat St000854

About

Examples	5
Crosswalk	FindStat