5.4 Fundamental weights and the Weyl group

Fundamental weights and the Weyl group are key players in understanding root systems and Dynkin diagrams. They help us navigate the complex world of Lie algebras by providing a framework for classifying representations and analyzing their properties.

These concepts are crucial for grasping how root systems work. The Weyl group's action on weights and the use of fundamental weights in character formulas give us powerful tools for studying Lie algebra representations.

Fundamental Weights and Root Lattice

Definition and Properties of Fundamental Weights

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• Fundamental weights are a special set of vectors in the dual space of the Cartan subalgebra, denoted as ω₁, ω₂, ..., ωₗ, where ℓ is the rank of the Lie algebra
• The fundamental weights are defined by their inner product with the simple roots: ⟨ωᵢ, αⱼ⟩ = δᵢⱼ, where δᵢⱼ is the Kronecker delta which equals 1 if i = j, and 0 otherwise
• The set of fundamental weights forms a basis for the weight lattice, which is the dual lattice to the root lattice
• Every weight of a finite-dimensional representation can be expressed as a linear combination of fundamental weights with integer coefficients (ω₁ + 2ω₂ - ω₃)

Relation to Root Lattice and Weyl Vector

• The root lattice is the Z-span of the simple roots, while the weight lattice is the Z-span of the fundamental weights
• The Weyl vector, denoted as ρ, is the sum of all fundamental weights: ρ = ω₁ + ω₂ + ... + ωₗ
• The Weyl vector plays a crucial role in the Weyl character formula and representation theory (used in the numerator of the character formula)
• The fundamental weights are the dual basis to the simple coroots, which are defined as α̌ = 2α/⟨α,α⟩ for each simple root α
• The coroots play a crucial role in the classification of semisimple Lie algebras (used in the Cartan matrix and Dynkin diagrams)

Weyl Group Action on Root System

Definition and Properties of Weyl Group

• The Weyl group W is a finite group generated by reflections associated with the root system of a semisimple Lie algebra
• For each root α, there is a reflection σₐ that acts on the Cartan subalgebra and its dual space. The reflection σₐ is defined by σₐ(λ) = λ - ⟨λ, α⟩α for any weight λ
• The Weyl group acts on the root system by permuting the roots. This action preserves the inner product between roots and the angles between them
• The Weyl group is isomorphic to a subgroup of the permutation group of the roots. For example, in type Aₙ (sl(n+1)), the Weyl group is isomorphic to the symmetric group S_{n+1}

Weyl Chambers and Simple Transitive Action

• The action of the Weyl group divides the root system into Weyl chambers, which are fundamental domains for the action of W
• The dominant Weyl chamber is the one containing all positive roots (the chamber where all simple roots have non-negative coefficients)
• The Weyl group acts simply transitively on the set of Weyl chambers, meaning that for any two Weyl chambers, there is a unique element of W that maps one to the other
• The length of a Weyl group element, defined as the minimum number of simple reflections needed to express it, appears in the Weyl character formula and is related to the geometry of the root system

Weyl Character Formula for Characters

Statement and Components of the Formula

• The Weyl character formula is a powerful tool for computing the characters of finite-dimensional representations of semisimple Lie algebras
• For a finite-dimensional representation V with highest weight λ, the character of V is given by: ch(V)(μ) = ∑{w ∈ W} det(w) e^{w(λ+ρ)-ρ}(μ) / ∏{α>0} (e^{α/2}(μ) - e^{-α/2}(μ)), where μ is an element of the Cartan subalgebra, and the product is over all positive roots α
• The Weyl character formula involves a sum over the Weyl group W and a product over the positive roots
• The determinant term det(w) is ±1, depending on whether w is an even or odd permutation of the roots

Applications and Significance

• The numerator e^{w(λ+ρ)-ρ}(μ) involves the Weyl group action on the highest weight λ shifted by the Weyl vector ρ
• The denominator ∏_{α>0} (e^{α/2}(μ) - e^{-α/2}(μ)) is the Weyl denominator and is independent of the specific representation V
• The Weyl character formula can be used to derive the dimensions of representations and to study their decomposition into irreducible representations (finding the multiplicities of weight spaces)
• The Weyl denominator identity expresses the Weyl denominator as an alternating sum over the Weyl group. This identity is crucial in the proof of the Weyl character formula

Weyl Group, Fundamental Weights, and Representation Theory

Connections between Weyl Group and Weight Lattice

• The Weyl group, fundamental weights, and representation theory are intricately connected in the study of semisimple Lie algebras
• The action of the Weyl group on the weight lattice preserves the set of weights of any finite-dimensional representation. In particular, the Weyl group permutes the weights within each Weyl orbit
• The dominant integral weights, which are the non-negative integer combinations of fundamental weights, parametrize the finite-dimensional irreducible representations of a semisimple Lie algebra (highest weight modules)

Weyl Group Action on Characters and Representations

• The Weyl group acts on the characters of representations, and the Weyl character formula relates the character of a representation to its highest weight and the action of the Weyl group
• The study of the Weyl group, its action on the root and weight lattices, and its connection to representation theory is central to the classification and understanding of semisimple Lie algebras and their representations
• The Weyl group action on weights allows for the construction of weight diagrams and the study of weight multiplicities in representations (the number of linearly independent vectors of a given weight)
• The Weyl group also plays a role in the tensor product decomposition of representations and the construction of invariant tensors (Clebsch-Gordan coefficients and invariant forms)