Weyl character formula

The Weyl character formula expresses the character of an irreducible representation of a semisimple Lie algebra or group as a quotient of alternating sums over its Weyl group.

Weyl group: The finite reflection group generated by reflections across the root hyperplanes of a root system. Its signed action produces the alternating sums in both numerator and denominator.
Character of a representation: The trace-valued function assigning to each group element or Lie algebra element the trace of its action in a representation. The formula computes this trace function for irreducible representations.
Weyl dimension formula: A product formula giving the dimension of a finite-dimensional irreducible representation from its highest weight and positive roots. Evaluating the character at the identity yields this dimension formula.
Hermann Weyl: A German mathematician whose work shaped Lie theory, quantum mechanics, and the theory of representations. The character formula bears his name and became a central result of his representation theory.
Root system: A finite configuration of vectors encoding the reflections and combinatorial structure of a semisimple Lie algebra. Its roots determine the reflection group and the denominator of the formula.
Semisimple Lie algebra: A Lie algebra with no nonzero solvable ideal, whose finite-dimensional representations admit structural decomposition. The formula applies to irreducible representations of this class of Lie algebras.
Schur polynomial: A symmetric polynomial that is the character of an irreducible polynomial representation of a general linear group. For general linear groups, the Weyl character formula specializes to the Schur polynomial determinantal formula.
Élie Cartan: A French mathematician who developed foundational structure theory for Lie groups and Lie algebras. His classification and structural results supplied essential context for the formula's setting.
Weyl denominator formula: An identity expressing a product over positive roots as an alternating sum over the Weyl group. It supplies the denominator and the normalization underlying the character formula.
Highest-weight representation: A representation generated by a vector annihilated by all positive-root operators and acted on diagonally by the Cartan subalgebra. Its highest weight is the input that identifies the irreducible character.

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