Homogeneous Polynomials in Two Variables

A family of representations important in our analysis of the hydrogen atom consists of the representations of SU (2) on spaces of homogeneous polynomials. These representations play a major role in our classification of representations in Chapter 6. 

Recall from Section 2.2 that the complex vector space P - of homogeneous complex polynomials of degree n in two variables has dimension n +-1 and has a basis of the form x ,, xy , y . The action of the 

Note that because the action of SU 2) on is linear and invertible, the transformation R (g) preserves polynomial degree. These representations are related to the spin of elementary particles as we will see in Section 10.4 in particular, P corresponds to a particle of spin-n/2. Spin is a quality of particles that physicists introduced into their equations to model certain mysterious experimental results we will see in Chapter 10 that spin arises naturally from the spherical symmetry of space.) 

Let us calculate some of these s explicitly. Recall that each element of a -p  

For n = 2 we have the three-dimensional vector space spanned by the basis a 2, a v, y . We have 

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Recall the vector space of homogeneous polynomials in two variables defined in Section 2.2. The vector space V is the tensor product ofP and P, denoted P P. In other words, the elements of V are precisely the linear combinations of terms of the form p(u, v)g (x, y), where is a homogeneous polynomial of degree one and q is a homogeneous polynomial of degree two. Note that, given an element r(M, u, x, y) of P 0p2 there are many different ways to write it as a linear combination of products. For example,... 

Exercise 2.1 Consider the set of homogeneous polynomials in two variables with real coefficients. There is a natural addition of polynomials and a natural scalar multiplication of a polynomial by a complex number. Show that the set of homogeneous polynomials with these two operations is not a complex vector space. 

We would like to find the character of each representation of 5(7(2) on homogeneous polynomials in two variables, introduced in Section 4.6. For each nonnegative integer n it suffices to find the diagonal entries of the matrix form of the transformation R g) in the familiar basis. We calculated some of... 

In other words, the representations U of 5m(2) as differential operators on homogeneous polynomials in two variables are essentially the only finitedimensional irreducible representations, and they are classified by their dimensions. Unlike the Lie group 50(3), the Lie algebra sm(2) has infinitedimensional irreducible representations on complex scalar product spaces. See Exercise 8.10. 

Recall the vector space P" of homogeneous polynomials of degree n in two variables defined in Section 2.2. We will find it useful (see Proposition 4.7) to define the following complex scalar product on P ... 

Figure 2.3. A picture of the basis of homogeneous polynomials of degree two in three variables.

We will reserve the plainer symbol P for homogeneous polynomials of degree n in only two variables, a star player in our drama.) The set P is a complex vector space of dimension 1 the set containing only the function f C, (x, y, z) 1 is a basis. Let T denote the restriction of the... 

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