The Weyl Character Formula
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چکیده
We have seen that irreducible representations of a compact Lie group G can be constructed starting from a highest weight space and applying negative roots to a highest weight vector. One crucial thing that this construction does not easily tell us is what the character of this irreducible representation will be. The character would tell us not just which weights occur in the representation, but with what multiplicities they occur (this multiplicity is one for the highest weight, but in general can be larger). Knowing the characters of the irreducibles, we can use this to compute the decomposition of an arbitrary representation into irreducibles. The character of a representation (π, V ) is the complex-valued, conjugationinvariant function on G given by
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