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

[lion8172]

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?

 

[CompuChip]

By: "the converse", you mean: If the highest weight of a representation is unique, it is irreducible?

 

[lion8172]

Yes.