Is Showing the Uniqueness of Highest Weight a Valid Proof of Irreducibility?

lion8172
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I know that the highest weight of an irreducible representation is unique (up to scaling). Is the converse true, however? I have seen some proofs of the irreducibility of certain representations that proceed by showing the uniqueness of the highest weight. How do we know that this is a valid approach?
 
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By: "the converse", you mean: If the highest weight of a representation is unique, it is irreducible?
 
Yes.
 

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