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Concept
Weyl character formula
Summary
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. It was proved by . There is a closely related formula for the character of an irreducible representation of a semisimple Lie algebra. 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. 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.
The character formula can be expressed in terms of representations of complex semisimple Lie algebras or in terms of the (essentially equivalent) representation theory of compact Lie groups.
Let be an irreducible, finite-dimensional representation of a complex semisimple Lie algebra . Suppose is a Cartan subalgebra of . The character of is then the function defined by
The value of the character at is the dimension of . By elementary considerations, the character may be computed as
where the sum ranges over all the weights of and where is the multiplicity of . (The preceding expression is sometimes taken as the definition of the character.)
The character formula states that may also be computed as
where
is the Weyl group;
is the set of the positive roots of the root system ;
is the half-sum of the positive roots, often called the Weyl vector;
is the highest weight of the irreducible representation ;
is the determinant of the action of on the Cartan subalgebra .
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