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 of a representation of G is the trace of , as a function of a group element . 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 of gives a lot of information about itself.
Weyl's formula is a closed formula for the character , in terms of other objects constructed from G and its Lie algebra.
^Hall 2015 Section 12.4.
^Hall 2015 Section 10.4.
^Hall 2015 Section 12.5.
and 22 Related for: Weyl character formula information
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