Weyl Group

The Weyl group of a root system is the finite group generated by reflections across the hyperplanes orthogonal to the roots. Named after Hermann Weyl, who recognized its central role in the representation theory of Lie groups, the Weyl group is the symmetry group of the root system itself — and therefore, by the classification of Wilhelm Killing and Élie Cartan, the symmetry group of the associated simple Lie algebra. It is not an auxiliary structure but a core actor: the geometry of the Weyl group governs the classification of representations, the computation of characters, and the combinatorial topology of flag varieties.

Structure and Generation

Given a root system Φ in a Euclidean space, the Weyl group W is generated by the simple reflections s_α across the hyperplanes orthogonal to the simple roots α. These generators satisfy the Coxeter relations: each reflection has order two, and the product of two reflections s_α s_β has order m_{αβ}, where m_{αβ} depends on the angle between the roots. The resulting presentation makes every Weyl group a Coxeter Group, and the classification of finite Coxeter groups coincides with the classification of root systems — the same Dynkin diagrams that classify simple Lie algebras also classify finite reflection groups.

The Weyl group acts simply transitively on the set of Weyl chambers, the connected components of the complement of the reflecting hyperplanes. This geometric fact has deep consequences: every element of the Weyl group can be written as a reduced word in the simple reflections, and the length of such a word — the number of reflections needed — corresponds to the number of positive roots sent to negative roots by that element. The Bruhat Order, a partial order on the Weyl group defined by this length function, becomes the combinatorial skeleton of the Schubert calculus on Flag Variety and governs the decomposition of algebraic varieties into cells.

The Weyl Group in Representation Theory

The representation theory of a semisimple Lie algebra is determined by the action of its Weyl group on the weight lattice. The highest weight theorem states that every finite-dimensional irreducible representation is uniquely determined by its highest weight, a dominant weight that is fixed by no non-identity element of the Weyl group. The Weyl Character Formula expresses the character of a representation as a ratio of alternating sums over the Weyl group — a formula of astonishing economy, in which the entire infinite-dimensional representation theory is compressed into a finite sum over the symmetry group of the root system.

The Weyl dimension formula, a corollary of the character formula, computes the dimension of the representation with highest weight λ as a product over the positive roots. The formula reveals that the dimension grows polynomially with the weight, and the degree of the polynomial is the number of positive roots. This is not a coincidence: the Weyl group is the symmetry that constrains the representation theory, and its order appears in the denominator of the character formula as a normalization factor that ensures the characters are orthonormal.

Classification and the Exceptional Cases

The Weyl groups of the classical Lie algebras — A_n, B_n, C_n, and D_n — are the symmetric group S_{n+1} and the hyperoctahedral groups. These are familiar combinatorial objects, and their representation theory is well understood. The exceptional Weyl groups — those of G₂, F₄, E₆, E₇, and E₈ — are more exotic. The Weyl group of E₈ has order 696,729,600, and its structure is so intricate that it has been studied as a mathematical object in its own right, with connections to the Monster Group and the theory of finite simple groups.

The exceptional Weyl groups resist classical geometric intuition. They do not act naturally on Euclidean spaces in a way that is easy to visualize, yet they govern the most symmetric structures in mathematics. The fact that the same finite list of reflection groups appears in Lie theory, singularity theory, and the combinatorics of convex polytopes is one of the deepest regularities in mathematics — a regularity that we can describe but not yet explain.

The Weyl group is not a mere symmetry of the root system; it is the reason the root system has a structure worth studying. Without the Weyl group, the roots are just vectors. With it, they become a dictionary for the representation theory of the universe's most symmetric objects. The classification of Weyl groups is not a taxonomic achievement — it is a proof that the space of symmetries is finite, closed, and complete. And that finiteness, that closure, is the most surprising fact in all of mathematics.