the above formulas and checking that the commutation relations hold. This representation can then be shown to be irreducible. Conclusion: For each non-negative integer there is a unique irreducible representation with highest weight Each irreducible representation is equivalent to one of these. The representation with highest weight has dimension with weights each having multiplicity one. The Casimir element We now introduce the (quadratic) Casimir element, given by . We can view as an element of the universal enveloping algebra or as an operator in each irreducible representation. Viewing as an operator on the representation with highest weight , we may easily compute that commutes with each Thus, by Schur's lemma, acts as a scalar multiple of the identity for each We can write in terms of the basis as follows: which can be reduced to The eigenvalue of in the representation with highest weight can be computed by applying to the highest weight vector, which is annihilated by thus, we get In the physics literature, the Casimir is normalized as Labeling things in terms of the eigenvalue of is then computed as The group representations Action on polynomials Since SU(2) is simply connected, a general result shows that every representation of its (complexified) Lie algebra gives rise to a representation of SU(2) itself. It is desirable, however, to give an explicit realization of the representations at the group level. The group representations can be realized on spaces of polynomials in two complex variables. That is, for each non-negative integer , we let denote the space of homogeneous polynomials of degree in two complex variables. Then the dimension of is . There is a natural action of SU(2) on each , given by . The associated Lie algebra representation is simply the one described in the previous section. (See here for an explicit formula for the action of the Lie algebra on the space of polynomials.) The characters The character of a representation is the function given by . Characters plays an important role in the representation theory of compact groups. The character is easily seen to be a class function, that is, invariant under conjugation. In the SU(2) case, the fact that the character is a class function means it is determined by its value on the maximal torus consisting of the diagonal matrices in SU(2), since the elements are orthogonally diagonalizable with the spectral theorem. Since the irreducible representation