Homogeneous polynomial

If the ligands are not allowed to be chiral, they can be characterized by a single scalar parameter A. The starting function y> in Eq. (1) is chosen to be a homogeneous polynomial in the A. Since each of the monomial terms in y> separately yields either zero or a function with the desired property, we can consider y> to be a monomial without loss of generality. We choose the monomial to be of the lowest order which will not be annihilated by the operations of Eq. (1). 

Homogeneous polynomials play an important role in our story. 

The right-hand side of each equation is a homogeneous polynomial of de-... 

Let us calculate, for future reference, the dimension of the complex vector space of homogeneous polynomials (with complex coefficients) of degree n on various Euclidean spaces. Homogeneous polynomials of degree n on the real line R are particularly simple. This complex vector space is onedimensional for each n. In fact, every element has the form ex for some c e C. In other words, the one-element set x" is a finite basis for the homogeneous polynomials of degree n on the real line. 

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... 

Recall from Section 1.5 that any function in the kernel of the Laplacian (on any space of functions) is called a harmonic function. In other words, a function f is harmonic if V / = 0. The harmonic functions in the example just above are the harmonic homogeneous polynomials of degree two. We call this vector space In Exercise 2,23 we invite the reader to check that the following set is a basis of H/ ... 

For example, there is an isomorphism from to the vector space P3 of homogeneous polynomials of degree one in three variables (x, y, z), given by... 

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. 

Exercise 2.9 (Relevant to Proposition 7.3) Suppose V is the vector space of all polynomials in three variables. Suppose q is a polynomial in three variables. Show that multiplication by q is a linear transformation. In other words, consider the function taking any p(x, y,f)eV to qix, y, z) p x, y, z). Show that this function is linear. What is its range Remark these statements hold true for polynomials in any number of variables.) Now let denote the homogeneous polynomials in three variables of degree I. Let r denote the polynomial + y + z . Show that r" ... 

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 ... 

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. 

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... 

Exercise 5.7 Recall the representations R of SU (2) on homogeneous polynomials introduced in Section 4.6. Find a complex scalar product on the vector space of the representation 7 i 7 2 such that the representation is unitary. Consider the subspace Vi spanned by uxy — vx, uy — rxy and the subspace Vj spanned by [ux", 2uxy + vx, 2vxy + uy, vy". Use this complex scalar product to find Is your answer isomorphic to V- Is it equal to V3 ... 

Exercise 5.8 Recall the representations R, of SU (2) on homogeneous polynomials introduced in Section 4.6. Check that the representation on the subspace V3 (defined in Exercise 5.7) is isomorphic to the representation R3 on V. Use the suggested isomorphism, and check that it satisfies Definition 4.9. It suffices to check that RjoTl p) = Top(p) for each of the four basis vectors p. (Here p is the representation on V3 and T is the alleged isomorphism.)... 

As an example of Proposition 6.8, consider the characters xo and xi of the representations of Sf/ 2 on the spaces of constant and degree-one (respectively) homogeneous polynomials of two variables. The proposition implies that (xo, = 0. We can check this result by direct calculation using the formulas from Section 4.6 and Equation 6.2 we have... 

Proof. Consider the vector spaces of homogeneous polynomials of degree (. and P3 of homogeneous polynomials of degree f — 2 in three variables. (Sticklers for rigor should define Pf. = Pf .= 0. ) Let denote the restriction of the Laplacian = -f 9 -j- to P. By I xercise 2.21 we know that the image of the linear transformation Vf lies in P. ... 

The surjectivity of the restricted Laplacian allows us to finish our computation of the dimension of the vector space of homogeneous harmoiuc polynomials of degree I. We already knew that the dimension of the domain of the restricted Laplacian was - - l)(f + 2). We now know that the dimension of the image of the restricted Laplacian is the dimension of that is, — l)f. Hence by Proposition 2.5 the dimension of the space of harmoiuc homogeneous polynomials of degree f is... 

Finally, since any polynomial is a finite sum of homogeneous polynomials, we conclude that ([Pg.216]

Each operator preserves the degree of homogeneous polynomials. For ex-... 

Hence Ui, Uj and Uk preserve the degree of any monomial. Hence U preserved the degree of any polynomial and takes any homogeneous polynomial to another homogeneous polynomial. In other words, each space P of homogeneous polynomials of a particular degree n is a subrepresentation of (5/(2), P, U). 

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. 

Suppose we have homogeneous polynomials. .., fs k[x0,. .., xr] of degrees d1(.., d5 respectively defining a closed subscheme X c Pr Then, in this section, we will refine some of the results of section 8 by investigating those infinitesimal deformations of X in Pr, parametrized by a k-algebra A in a, which can be defined by s homogeneous polynomials. .., Fs t A[x] whose reductions mod m.A are fi, respectively... 

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