HighestRootWeylVectorPairing

The symmetric bilinear form of the highest root with the Weyl vector, normalized so that the shortest roots have squared length 2.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001678via CartanType✓↗

HighestRootWeylVectorPairing(cartanType)The symmetric bilinear form of the highest root with the Weyl vector, normalized so that the shortest roots have squared length 2.

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

Apply(
       (ds: list<integer>) => _x[2] if _x[1] == "A" else 4 * _x[2] - 4 if _x[1] ==
                                                                          "B" else 2 * _x[2] if _x[1] == "C" else 2 * _x[2] -
                                                                                                               3 if _x[1] == "D" else Max(ds) - 1 if _x[1] == "E" else 16 if _x[1] == "F" else 9,
       Range(2, _x[2] + 1, 1) if _x[1] == "A" else Map(
                                                        Function(2i, i),
                                                        Range(1, _x[2]),
                                                      ) if _x[1] == "B" else Map(
                                                                                 Function(2i, i),
                                                                                 Range(1, _x[2]),
                                                                               ) if _x[1] == "C" else Join(
                                                                                                           Filter(
                                                                                                                   Map(Function(2i, i), Range(1, _x[2] - 1)),
                                                                                                                   Function(e < _x[2], e),
                                                                                                                 ),
                                                                                                           [_x[2]],
                                                                                                           Filter(
                                                                                                                   Map(Function(2i, i), Range(1, _x[2] - 1)),
                                                                                                                   Function(e >= _x[2], e),
                                                                                                                 ),
                                                                                                         ) if _x[1] == "D" else ([2, 5, 6, 8, 9, 12] if _x[2] == 6 else [2, 6, 8, 10, 12, 14, 18] if _x[2] == 7 else [2, 8, 12, 14, 18, 20, 24, 30]) if _x[1] == "E" else [2, 6, 8, 12] if _x[1] == "F" else [2, 6],
     )

Examples

Basic #

the type $D_5$

live ↗

In$\mathrm{HighestRootWeylVectorPairing}(\operatorname{CartanType}((\text{D},5)))$Out$7$

the type $E_6$

live ↗

In$\mathrm{HighestRootWeylVectorPairing}(\operatorname{CartanType}((\text{E},6)))$Out$11$

Properties #

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

live ↗

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

Known value · FindStat St001678

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

live ↗

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

Known value · FindStat St001678

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

live ↗

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

Known value · FindStat St001678

About

Examples	5
Crosswalk	FindStat