QuasiMinusculeRepresentationDimension

The dimension of the quasi-minuscule representation, whose highest weight is the highest short root.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001149via CartanType✓↗

QuasiMinusculeRepresentationDimension(cartanType)The dimension of the quasi-minuscule representation, whose highest weight is the highest short root.

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] * (_x[2] + 2) if _x[1] == "A" else 2 * _x[2] + 1 if _x[1] == "B" else 2 *
                                                                            _x[2] ^
                                                                            2 -
                                                                            _x[2] - 1 if _x[1] == "C" else _x[2] *
                                                                                                      (2 * _x[2] -
                                                                                                       1) if _x[1] == "D" else (78 if _x[2] == 6 else 133 if _x[2] == 7 else 248) if _x[1] == "E" else 26 if _x[1] == "F" else 7

Examples

Basic #

the type $D_5$

live ↗

In$\mathrm{QuasiMinusculeRepresentationDimension}(\operatorname{CartanType}((\text{D},5)))$Out$45$

the type $E_6$

live ↗

In$\mathrm{QuasiMinusculeRepresentationDimension}(\operatorname{CartanType}((\text{E},6)))$Out$78$

Properties #

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

live ↗

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

Known value · FindStat St001149

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

live ↗

In$\mathrm{CombinatorialStat}(\operatorname{CartanType}(("A",2)), \text{QuasiMinusculeRepresentationDimension})$Out$8$

Known value · FindStat St001149

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

live ↗

In$\mathrm{CombinatorialStat}(\operatorname{CartanType}(("A",3)), \text{QuasiMinusculeRepresentationDimension})$Out$15$

Known value · FindStat St001149

About

Examples	5
Crosswalk	FindStat