Struct E8RootSystem 

pub struct E8RootSystem { /* private fields */ }

E₈ Root System

Encapsulates all 240 roots of E₈ with exact arithmetic.

Invariants

• Exactly 240 roots total
• Exactly 112 integer roots
• Exactly 128 half-integer roots
• Every root has norm² = 2
• Every root r has negation -r in the system

Implementations

impl E8RootSystem

pub fn new() -> Self

Create new E₈ root system

Generates all 240 roots and verifies invariants.

pub const fn num_roots(&self) -> usize

Get total number of roots

pub fn get_root(&self, index: usize) -> &Vector8

Get root by index

Panics

Panics if index is out of bounds

pub fn get_negation(&self, index: usize) -> usize

Get index of negation of a root

Panics

Panics if index is out of bounds

pub fn sign_class_representative(&self, index: usize) -> usize

Get sign class representative (smaller of {r, -r})

pub fn count_sign_classes(&self, indices: &[usize]) -> usize

Count number of distinct sign classes used by a set of root indices

pub fn roots(&self) -> &[Vector8]

Get all roots as a slice

pub fn inner_product(&self, i: usize, j: usize) -> Rational

Compute inner product between two roots

pub fn are_negatives(&self, i: usize, j: usize) -> bool

Check if two roots are negatives of each other

Returns true if root j is the negative of root i

pub fn find_root(&self, root: &Vector8) -> Option<usize>

Find index of a given root vector

Returns None if root is not in the system.

pub const fn simple_roots() -> [Vector8; 8]

Get the 8 simple roots of E₈

Simple Roots

The simple roots form a basis for the root system, with special properties:

• Every positive root is a non-negative integer linear combination of simple roots
• They form a linearly independent set spanning the root space
• The Weyl group is generated by reflections through these roots

For E₈, we use the standard simple root basis:

• α₁ = e₁ - e₂ = (1, -1, 0, 0, 0, 0, 0, 0)
• α₂ = e₂ - e₃ = (0, 1, -1, 0, 0, 0, 0, 0)
• α₃ = e₃ - e₄ = (0, 0, 1, -1, 0, 0, 0, 0)
• α₄ = e₄ - e₅ = (0, 0, 0, 1, -1, 0, 0, 0)
• α₅ = e₅ - e₆ = (0, 0, 0, 0, 1, -1, 0, 0)
• α₆ = e₆ - e₇ = (0, 0, 0, 0, 0, 1, -1, 0)
• α₇ = e₇ + e₈ = (0, 0, 0, 0, 0, 0, 1, 1)
• α₈ = ½(-1, -1, -1, -1, -1, -1, -1, -1)

This basis encodes the E₈ Dynkin diagram structure:

• Roots α₁ through α₇ form an A₇ chain
• Root α₈ connects to α₇, creating the E₈ branching

Returns

Array of 8 simple roots with norm² = 2

Trait Implementations

impl Clone for E8RootSystem

fn clone(&self) -> E8RootSystem

Returns a duplicate of the value.

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

Performs copy-assignment from source.

impl Debug for E8RootSystem

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

Formats the value using the given formatter.

impl Default for E8RootSystem

fn default() -> Self

Returns the “default value” for a type.