Function, symmetric

Among other things, Redfield s paper led to a heightened awareness of something that was already beginning to be realized, namely the interrelationship between Polya s Theorem (and other enumeration theorems) on the one hand, and the theory of symmetric functions, -functions, and group characters on the other it helped to show the way to the use of cycle index sums in the solution of hitherto intractable problems and in a more nebulous way it provided a refreshing new outlook on combinatorial problems. 

The light shed by Redfield s paper on the close connection between Polya theory and symmetric function theory is well illustrated by a particularly simple way of looking at Polya s Theorem -- one that shows the way to further developments. Suppose the store of figures consists of n distinct figures, as for example with necklace problems using n kinds of beads. The figure generating function is then... 

But this latter expression, called a power sum, is of common occurrence in the theory of symmetric functions, where it is universally denoted by s. It was for this reason that was used in the notation for the cycle index rather than Pdlya s... 

It was shown in [ReaR68] that the operation N A B) which is required by the superposition theorem is particularly simple if the operands A and B are the symmetric functions known as -functions (or Schur functions). In fact, for any two -functions X and /z ... 

The continuum model with the Hamiltonian equal to the sum of Eq. (3.10) and Eq. (3.12), describing the interaction of electrons close to the Fermi surface with the optical phonons, is called the Takayama-Lin-Liu-Maki (TLM) model [5, 6], The Hamiltonian of the continuum model retains the important symmetries of the discrete Hamiltonian Eq. (3.2). In particular, the spectrum of the single-particle states of the TLM model is a symmetric function of energy. 

For nondegenerate vibrations all symmetry operations change Qj into 1 times itself. Hence Q/ is unchanged by all symmetry operations. In other words, Q and consequently y(O) behave as totally symmetric functions (i.e. the function is independent of symmetry). However, the wavefunction of the first excited state 3(1) has the same symmetry as Qj. For example, the wavefunction of a totally symmetric vibration (e.g. Qi of C02) is itself a totally symmetric function. 

The radial frequency co of a periodic function is positive or negative, depending on the direction of the rotation of the unit vector (see Fig. 40.5). co is positive in the counter-clockwise direction and negative in the clockwise direction. From Fig. 40.5a one can see that the amplitudes (A jp) of a sine at a negative frequency, -co, with an amplitude. A, are opposite to the values of a sine function at a positive frequency, co, i.e. = Asin(-cor) = -Asin(co/) = This is a property of an antisymmetric function. A cosine function is a symmetric function because A -Acos(-co/) = Acos(cor) = A. (Fig. 40.5b). Thus, positive as well as negative... 

The elastic free energy given by the elementary and the more advanced theories are symmetric functions of the three extension ratios Xx, Xy, and Xz. One may also express the dependence of the elastic free energy on strain in terms of three other variables, which are in turn functions of Xx, Xy, and Xz. In phenomenological theories of continuum mechanics, where only the observed behavior of the material is of concern rather than the associated molecular deformation mechanisms, these three functions are chosen as... 

In the deduction Ruland determines, which contribution to the observed intensity, /, is added by each reflection ring of the likewise fiber symmetrical function, Iopt. Then he adds up all the rings weighted by the orientation function g 3). In this way Eq. (9.11) is simplified. A general solution is obtainable by multipole expansion. 

In essence, it is the probability density of the two nuclei to have relative separation Since the orientation of the molecule is not fixed (nuclei are not fixed any more if we deal with an non-BO approach), gi( ) is a spherically symmetric function. The plots of gi ( ) are presented in Figs. 1-4-. It should be noted that all the correlation functions shown are normalized in such a way that... 

For even indices, Unix) is an even function, therefore the transmittance is a symmetric function of energy. For u=l, r is constant and equals unity by definition, as we have mentioned. When n—2, we obtain ... 

Symmetric products of an embedded curve, symmetric functions... 

This chapter is continuation of the previous chapter. We shall study the operator PpiM associated with an embedded curve E in X. There are also distinguished homology classes [L E] in //h<(A N) (n = iy ) introduced in Chapter 7. The operator P[s][ ] preserves these classes. We shall give the precise formulas for the action. We shall find they are exactly the same as relations in symmetric functions when we replace P[s][ ] by the power sump, [L E] by the orbit sum rrij,. 

The idea to use symmetric products of an embedded curve is due to Grojnowski [33], although the relation to symmetric functions seems to new. The relation to vertex algebras is also due to him. 

First we briefly recall the theory of symmetric functions which will be needed in the next section. See [54] for detail. 

Let Aat be the right of symmetric functions with rational coefficients... 

In the relationship between symmetric functions and Hilbert schemes, the degree n corresponds the number of points, while the number of variables N are irrelevant. This is the only reason why we use the different notation from [54]. 

There are several distinguished classes of symmetric functions. The first class is the orbit sum rrii,. Let be a partition with l y) < N. Let... 

For each nonnegative integer n, let us dehne the n-th elementary symmetric function 6n by... 

SYMMETRIC PRODUCTS, SYMMETRIC FUNCTIONS AND VERTEX OPERATORS... 

Although it is not related to the rest of this chapter, it is worth while to remark that the relationship between Chern classes/characters and symmetric functions. If i is a complex vector bundle of rank N, we can define its Ath Chern classes q. The total Chern class is its generating function ... 

A symmetric function in Xi, which is a polynomial in Cd Eys, defines a well-defined cohomology class in H X). Though c E) is not necessarily factored in H X), we could think as if Xi s are well-defined classes when we want to prove a universal formula involving Chern classes. This follows from the splitting principle. For example, we could define the Chern character by... 

---