- Weyl character formula
mathematics, the Weyl character formula in representation theorydescribes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It is named after Hermann Weyl, who proved it in the late 1920s.
By definition, the character of a representation "r" of "G" is the trace of "r"("g"), as a function of a group element "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 χ of "r" is a good substitute for "r" itself, and can have algorithmic content. Weyl's formula is a closed formulafor the χ, in terms of other objects constructed from "G" and its Lie algebra. The representations in question here are complex, and so without loss of generality are unitary representations; "irreducible" therefore means the same as "indecomposable", i.e. not a direct sum of two subrepresentations.
tatement of Weyl character formula
The character of an
irreducible representation"V" of a compact Lie group "G" is given by
*ρ is the Weyl vector of the group "G", defined to be half the sum of the positive roots;
*"W" is the
*λ is the
highest weightof the irreducible representation "V";
*α runs over the
positive roots of the Lie group.
Weyl denominator formula
In the special case of the trivial 1 dimensional representation the character is 1, so the Weyl character formula becomes the Weyl denominator formula:
For special unitary groups, this is equivalent to the expression :
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