Struct Alcove 

pub struct Alcove { /* private fields */ }

Fundamental alcove for an affine root system.

An alcove is a bounded region in the Cartan space defined by affine hyperplanes (root hyperplanes shifted by integer levels). The fundamental alcove is:

A = {λ ∈ 𝔥* : ⟨λ, αᵢ⟩ ≥ 0 for all simple αᵢ, and ⟨λ, θ⟩ ≤ 1}

where θ is the highest root (longest root in the positive system).

Mathematical Background

• Alcoves tile 𝔥* under the affine Weyl group Wₐff
• The fundamental alcove parametrizes integrable highest-weight modules at level k (conformal field theory, loop groups)
• Vertices of the alcove are fundamental weights

Example

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

let root_system = RootSystem::type_a(2); // SU(3)
let alcove = Alcove::fundamental(&root_system);

// Weight inside alcove
let lambda = Root::new(vec![0.3, 0.3, -0.6]);
// (need to verify ⟨λ, αᵢ⟩ ≥ 0 and ⟨λ, θ⟩ ≤ 1)

References

• Kac, Infinite Dimensional Lie Algebras, §6.2
• Humphreys, Reflection Groups and Coxeter Groups, §4.4

Implementations

impl Alcove

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

Construct the fundamental alcove at level k = 1.

The fundamental alcove is:

A = {λ : ⟨λ, αᵢ⟩ ≥ 0, ⟨λ, θ⟩ ≤ 1}

pub fn at_level(root_system: &RootSystem, level: f64) -> Self

Construct an alcove at a specified level k.

Arguments

• root_system - The root system
• level - Affine level k (positive integer for integrable modules)

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

Check if a weight is in the alcove.

Arguments

• weight - Weight λ ∈ 𝔥* to test
• strict - If true, use strict inequalities

Returns

True if λ is in the alcove

pub fn level(&self) -> f64

Get the level of the alcove.

pub fn highest_root(&self) -> &Root

Get the highest root defining the upper wall.

pub fn vertices(&self) -> Vec<Root>

Compute vertices of the fundamental alcove.

For type A_n, the fundamental alcove is a simplex with n+1 vertices: the origin and the fundamental weights {ω₁, …, ωₙ}.

Returns

Vector of vertices (as Roots in 𝔥*)

Trait Implementations

impl Clone for Alcove

fn clone(&self) -> Alcove

Returns a duplicate of the value.

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

Performs copy-assignment from source.

impl Debug for Alcove

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

Formats the value using the given formatter.