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)
Famous quotes containing the word weights:
“What do you believe in?—In this, that the weights of all things must be determined anew.”
—Friedrich Nietzsche (1844–1900)
“This is essentially a People’s contest. On the side of the Union, it is a struggle for maintaining in the world, that form, and substance of government, whose leading object is, to elevate the condition of men—to lift artificial weights from all shoulders—to clear the paths of laudable pursuit for all—to afford all, an unfettered start, and a fair chance, in the race of life.”
—Abraham Lincoln (1809–1865)
“... the prevalent custom of educating young women only for marriage, and not for the duties and responsibilities consequent on marriage—only for appendages and dead weights to husbands—of bringing them up without an occupation, profession, or employment, and thus leaving them dependent on anyone but themselves—is an enormous evil, and an unpardonable sin.”
—Harriot K. Hunt (1805–1875)