Weyl character formula

In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights.^[1] It was proved by Hermann Weyl (1925, 1926a, 1926b). There is a closely related formula for the character of an irreducible representation of a semisimple Lie algebra.^[2] In Weyl's approach to the representation theory of connected compact Lie groups, the proof of the character formula is a key step in proving that every dominant integral element actually arises as the highest weight of some irreducible representation.^[3] Important consequences of the character formula are the Weyl dimension formula and the Kostant multiplicity formula.

By definition, the character ${\displaystyle \chi }$ of a representation ${\displaystyle \pi }$ of G is the trace of ${\displaystyle \pi (g)}$, as a function of a group element ${\displaystyle g\in G}$. The irreducible representations in this case are all finite-dimensional (this is part of the Peter–Weyl theorem); so the notion of trace is the usual one from linear algebra. Knowledge of the character ${\displaystyle \chi }$ of ${\displaystyle \pi }$ gives a lot of information about ${\displaystyle \pi }$ itself.

Weyl's formula is a closed formula for the character ${\displaystyle \chi }$, in terms of other objects constructed from G and its Lie algebra.