Weyl character formula

Weyl character formula is a central result in the representation theory of compact Lie groups and complex semisimple Lie algebras that expresses the character of an irreducible highest-weight representation as an explicit ratio involving sums over a Weyl group and a product over roots. The formula connects concrete algebraic data—weights, roots, Weyl group elements—with classical objects studied by Hermann Weyl, Élie Cartan, Élie Joseph Cartan contemporaries and later contributors such as Claude Chevalley and Harish-Chandra. It plays a foundational role in the classification of representations for groups like SU(n), SO(n), and Sp(n) and in the work of mathematicians such as Hermann Weyl, Robert Steinberg, George Lusztig, and Bertram Kostant.

Statement

For a complex semisimple Lie algebra associated with a compact Lie group such as SU(n), SO(n), or Sp(n), let λ denote a dominant integral highest weight and let ρ denote the half-sum of positive roots defined by choices made by Élie Cartan and later systematized by Nathan Jacobson. The character χ_λ of the irreducible representation V(λ) is given by an equality of formal sums on the weight lattice: the numerator is a signed sum over the Weyl group W (studied in the context of Hermann Weyl and Élie Cartan), while the denominator is the Weyl denominator — a product indexed by the positive roots determined by the root system classified by Eugenio Elia Levi-Civita historically and formalized in the work of Élie Cartan and Élie Joseph Cartan. The resulting expression is invariant under W and yields explicit multiplicities via expansion methods used by Bertram Kostant and Harish-Chandra.

Background and prerequisites

Understanding the formula requires familiarity with structures developed by Élie Cartan, Hermann Weyl, Élie Joseph Cartan, and later algebraists: Cartan subalgebras, root systems and Dynkin diagrams classified by Wilhelm Killing and Élie Cartan, weight lattices and integral weights studied in the work of Claude Chevalley and H. Weyl, and Weyl groups arising in the theory of reflection groups related to James Joseph Sylvester and Arthur Cayley. One must know the construction of highest-weight modules from the perspective of Élie Cartan and Harish-Chandra and the role of the Borel subalgebra as in the work of Élie Joseph Cartan and Cartan–Weyl theory. Familiarity with characters as class functions on compact Lie groups treated by Hermann Weyl and harmonic analysis on compact groups as in the work of Norbert Wiener and Einar Hille is helpful. Algebraic and combinatorial tools developed by George Lusztig, Bertram Kostant, and Robert Steinberg—including the use of Verma modules and the Kostant partition function—are standard prerequisites.

Proofs and derivations

Classical derivations trace to arguments by Hermann Weyl using orthogonality relations for characters and integration over maximal tori, invoking results attributed to Élie Cartan and harmonic analysis on compact groups as developed by Harish-Chandra. Algebraic proofs use highest-weight theory, Verma modules introduced by I. N. Bernstein and Joseph Bernstein in related contexts, and the Shapovalov form studied by Nikita Nekrasov and others; these approaches were refined by Bertram Kostant who related multiplicities to the Kostant partition function and homological algebra techniques of Jean-Louis Koszul. Geometric proofs employ the Borel–Weil theorem linking representation spaces to holomorphic sections on flag varieties studied by Armand Borel and André Weil, with alternative derivations using localization in equivariant cohomology developed by Michael Atiyah and Raoul Bott. Modern categorical and quantum approaches link the formula to the work of Andrei Okounkov, George Lusztig, and research in geometric representation theory stemming from Alexander Beilinson and Joseph Bernstein.

Applications and consequences

The formula yields explicit weight multiplicities for irreducible representations of classical groups such as SU(n), SO(n), and Sp(n), enabling computations used in the representation theory of finite groups of Lie type studied by Robert Steinberg and in the theory of automorphic forms developed by Robert Langlands and Harish-Chandra. It underlies character tables in the work of Richard Brauer and Issai Schur and feeds into the study of tensor product decompositions related to the Littlewood–Richardson rule associated with Alfred Littlewood and D. E. Littlewood's collaborators. The formula informs the structure of affine Lie algebras analyzed by Victor Kac and the representation-theoretic side of the Toda lattice and integrable systems investigated by Mikhail Krichever and Lax Pair-related authors. In geometric representation theory, consequences appear in the study of Schubert calculus on flag varieties by Hermann Schubert and in modular representation theory developments by George Lusztig.

Examples and computations

For the classical series, explicit instances are standard: for SU(2), the Weyl group is of order two studied in early group theory by Arthur Cayley; the formula recovers the characters of spin j representations as trigonometric Chebyshev-type expressions historically compared in works of Hermann Weyl. For SU(n), weights correspond to partitions linked to the Young diagram combinatorics of Alfred Young and characters coincide with Schur polynomials appearing in the theory of Issai Schur and symmetric functions studied by I. G. Macdonald. Computations for SO(2n), SO(2n+1), and Sp(n) use root data classified in the Cartan–Killing classification by Wilhelm Killing and Élie Cartan; explicit multiplicities follow from expansion techniques due to Bertram Kostant and combinatorial rules developed by Andrei Zelevinsky and Richard Stanley. Advanced examples include applications to affine Lie algebras in the work of Victor Kac and modular invariant partition functions in conformal field theory where authors like Alexander Zamolodchikov and P. Di Francesco connect character formulas to modular transformations.

Category:Representation theory