Struct RootSystem 

pub struct RootSystem { /* private fields */ }

A root system for a semisimple Lie algebra.

Encodes the structure of the Lie bracket through roots and Weyl reflections.

Example

use lie_groups::RootSystem;

// SU(3) = type A₂
let su3 = RootSystem::type_a(2);
assert_eq!(su3.rank(), 2);
assert_eq!(su3.num_roots(), 6); // 3² - 1 = 8, but we store 6 roots
assert_eq!(su3.num_positive_roots(), 3);

Implementations

impl RootSystem

pub fn type_a(n: usize) -> Self

Create a type Aₙ root system (SU(n+1)).

For SU(n+1), the rank is n, and roots are differences eᵢ - eⱼ for i ≠ j. Simple roots: αᵢ = eᵢ - eᵢ₊₁ for i = 1, …, n.

Example

use lie_groups::RootSystem;

// SU(2) = A₁
let su2 = RootSystem::type_a(1);
assert_eq!(su2.rank(), 1);
assert_eq!(su2.num_roots(), 2); // ±α

// SU(3) = A₂
let su3 = RootSystem::type_a(2);
assert_eq!(su3.rank(), 2);
assert_eq!(su3.simple_roots().len(), 2);

pub fn rank(&self) -> usize

Rank of the Lie algebra.

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

All roots (positive and negative).

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

Simple roots (basis for root system).

pub fn cartan_matrix(&self) -> &[Vec<i32>]

Cartan matrix A_ij = ⟨α_j, α_i^∨⟩.

pub fn num_roots(&self) -> usize

Number of roots.

pub fn num_positive_roots(&self) -> usize

Number of positive roots.

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

Get all positive roots.

pub fn highest_root(&self) -> Root

Get the highest root (longest root in the positive system).

The highest root θ is the unique positive root with maximal height (sum of coefficients when expanded in simple roots). It’s also the longest root in the root system for simply-laced types like Aₙ.

For type A_n (SU(n+1)), the highest root is θ = α₁ + α₂ + … + αₙ.

Example

use lie_groups::RootSystem;

let su3 = RootSystem::type_a(2);
let theta = su3.highest_root();

// For SU(3), θ = α₁ + α₂ = (1, 0, -1)

pub fn contains_root(&self, root: &Root) -> bool

Check if a root is in the system.

pub fn weyl_reflection(&self, alpha: &Root, beta: &Root) -> Root

Weyl reflection s_α for a root α.

pub fn weyl_orbit(&self, weight: &Root) -> Vec<Root>

Generate the Weyl group orbit of a weight under simple reflections.

The Weyl group is generated by reflections in simple roots. For type Aₙ, |W| = (n+1)!

pub fn dimension(&self) -> usize

Dimension of the Lie algebra: rank + num_roots.

For type Aₙ: dim = n + n(n+1) = n(n+2)

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

Dominant weight chamber: λ such that ⟨λ, α⟩ ≥ 0 for all simple roots α.

pub fn simple_root_expansion(&self, root: &Root) -> Option<Vec<i32>>

Express a root as a linear combination of simple roots.

Returns coefficients [c₁, c₂, …, cₙ] such that β = Σ cᵢ αᵢ. For roots in the system, coefficients are integers (positive for positive roots).

Returns None if the root is not in this system, or if the expansion is not yet implemented for general root systems.

Supported Systems

• Type A (SU(n+1)): Fully implemented. Roots e_i - e_j expand as sums of consecutive simple roots.
• Other types: Returns None. General expansion requires Cartan matrix inversion, which is not yet implemented.

Trait Implementations

impl Clone for RootSystem

fn clone(&self) -> RootSystem

Returns a duplicate of the value.

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

Performs copy-assignment from source.

impl Debug for RootSystem

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

Formats the value using the given formatter.