Weights
A highest weight representation is a representation with a weight α which is greater than all the other weights.
If x is an eigenvector of α, e=0.
If the representation is irreducible,
and so, since x is nonzero, α is either λ or −λ−1.
A lowest weight representation is a representation with a weight α which is lower than all the other weights.
If x is an eigenvector of α, f=0.
If the rep is irreducible,
and so, α is either λ+1 or −λ.
Finite-dimensional representations only have finitely many weights, and so are both highest and lowest weight representations. For an irreducible finite-dimensional representation, the highest weight can't be less than the lowest weight. In addition, the difference between them has to be an integer because since
implies
and
implies
- .
If the difference isn't an integer, there will always be a weight which is one more or one less than any given weight, contradicting the assumption of finite dimensionality.
Since λ < λ+1 and −λ−1 < −λ, without any loss of generality we can assume the highest weight is λ (if it's −λ−1, just redefine a new λ’ as −λ−1) and the lowest weight would then have to be −λ. This means λ has to be an integer or half-integer. Every weight is a number between λ and −λ which differs from them by an integer and has multiplicity one. This can be seen by assuming otherwise. Then, we can define a proper subrepresentation generated by an eigenvector of λ and f applied to it any number of times, contradicting the assumption of irreducibility.
This construction also shows for any given nonnegative integer multiple of half λ, all finite dimensional irreps with λ as its highest weight are equivalent (just make an identification of a highest weight eigenvector of one with one of the other).
Read more about this topic: Representation Theory Of SU(2)
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