Struct WeylChamber 

pub struct WeylChamber { /* private fields */ }

Weyl chamber for a root system.

A Weyl chamber is a connected component of the complement of the reflecting hyperplanes in the dual Cartan space 𝔥*. The fundamental Weyl chamber (or dominant chamber) is defined by the simple roots:

C = {λ ∈ 𝔥* : ⟨λ, αᵢ⟩ ≥ 0 for all simple roots αᵢ}

Mathematical Background

• The Weyl group W acts on 𝔥* by reflections through root hyperplanes
• W partitions 𝔥* into finitely many Weyl chambers
• All chambers are isometric under W
• The fundamental chamber parametrizes dominant weights (highest weights of irreducible representations)

Example

use lie_groups::root_systems::{RootSystem, WeylChamber, Root};

let root_system = RootSystem::type_a(2); // SU(3)
let chamber = WeylChamber::fundamental(&root_system);

// Fundamental weight ω₁ = (2/3, -1/3, -1/3) is dominant (traceless for SU(3))
let omega1 = Root::new(vec![2.0/3.0, -1.0/3.0, -1.0/3.0]);
assert!(chamber.contains(&omega1, false)); // Non-strict since on boundary

// Negative weight is not dominant
let neg = Root::new(vec![-1.0, 0.0, 1.0]);
assert!(!chamber.contains(&neg, false));

References

• Humphreys, Introduction to Lie Algebras, §10.2
• Hall, Lie Groups, §8.4

Implementations

impl WeylChamber

pub fn fundamental(root_system: &RootSystem) -> Self

Construct the fundamental Weyl chamber for a root system.

The fundamental chamber is bounded by hyperplanes perpendicular to the simple roots:

C = {λ : ⟨λ, αᵢ⟩ ≥ 0 for all simple αᵢ}

pub fn contains(&self, weight: &Root, strict: bool) -> bool

Check if a weight is in the Weyl chamber.

Arguments

• weight - Weight λ ∈ 𝔥* to test
• strict - If true, use strict inequality (⟨λ, αᵢ⟩ > 0); else non-strict (≥ 0)

Returns

True if λ is in the chamber (dominant weight)

pub fn project(&self, weight: &Root) -> Root

Project a weight onto the closure of the fundamental Weyl chamber.

Uses the Weyl group to map an arbitrary weight to its unique dominant representative (closest point in the chamber).

Algorithm

Iteratively reflect through violated hyperplanes until all simple root pairings are non-negative.

pub fn simple_roots(&self) -> &[Root]

Get the simple roots defining the chamber walls.

Trait Implementations

impl Clone for WeylChamber

fn clone(&self) -> WeylChamber

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl Debug for WeylChamber

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.