Irreducible projective representation

In this section we define irreducible projective representations and find the irreducible projective representations of 50(3). These turn out to correspond to the different kinds of spin elementary particles can have, namely, 0, 1/2, 1, 3/2. 

All the irreducible linear Lie group representations of 5 U (2) correspond to spin representations of particles, i.e., to irreducible projective representations. The definition is quite natural. 

Proof of the Correspondence between Irreducible Linear Representations of 5f/(2) and Irreducible Projective Representations of 5<9(3)... 

Proof, (of Proposition 10.6) First we suppose that (S(/(2), V, p) is a linear irreducible unitary Lie group representation. By Proposition 6.14 we know that p is isomorphic to the representation R for some n. By Proposition 10.5 we know that R can be pushed forward to an irreducible projective representation of SO(3). Hence p can be pushed forward to an irreducible projective Lie group representation of SO(3). 

It is natural to wonder whether we have missed any irreducible projective unitary representations of 50(3). Are there any others besides those that come from irreducible linear representations The answer is no. 

Proposition 10.6 The irreducible projective unitaty representations of the Lie group SO if) are in one-to-one correspondence with the irreducible (linear) unitary representations of the Lie group SU (2). 

The results of this section are another confirmation of the philosophy spelled out in Section 6.2. We expect that the irreducible representations of the symmetry group determined by equivalent observers should correspond to the elementary systems. In fact, the experimentally observed spin properties of elementary particles correspond to irreducible projective unitary representations of the Lie group SO(3). Once again, we see that representation theory makes a testable physical prediction. 

It is recommended that the reader become familiar with the point-group symmetry tools developed in Appendix E before proceeding with this section. In particular, it is important to know how to label atomic orbitals as well as the various hybrids that can be formed from them according to the irreducible representations of the molecule s point group and how to construct symmetry adapted combinations of atomic, hybrid, and molecular orbitals using projection operator methods. If additional material on group theory is needed. Cotton s book on this subject is very good and provides many excellent chemical applications. 

More generally, it is possible to combine sets of Cartesian displacement coordinates qk into so-called symmetry adapted coordinates Qrj, where the index F labels the irreducible representation and j labels the particular combination of that symmetry. These symmetry adapted coordinates can be formed by applying the point group projection operators to the individual Cartesian displacement coordinates. 

An example of the application of Eq. (47) is provided by the group < 3v whose symmetry operations are defined by Eqs. (18). If the same arbitrary function,

symmetry operation can be worked out, as shown in the last column of Table 13. With the use of the projection operator defined by Eq. (47) and the character table (Table 6), it is found (problem 16) that the coordinate z is totally symmetric (representation Ai). However, it is the sum xy + zx that is preserved in the doubly degenerate representation, E. It should not be surprising that the functions xy and zx are projected as the sum, because it was the sum of the diagonal elements (the trace) of the irreducible representation that was employed in each case in the... 

D. Definitions of Young diagrams, tableaux, and operators should be understood, as well as the property of the Young operator of projecting onto an irreducible representation (Theorems 1 and 2). 

The irreducible representations of may be used to define a set of elements of the group algebra called "projection operators . The projection operator associated with the / th irreducible representation is defined by... 

For each irreducible representation JW there are different projection operators, so in all there are 2 n — g such operators. As we shall see presently, they are linearly independent, and thus form an alternative basis for the group algebra any element of U can be represented as a linear combination of the... 

As remarked above (cf. Eq. (9)), a representation of the group leads to a representation of the algebra in particular, each of the projection operators will be associated with a matrix. If the representation is the irreducible one I M, we find, using (9) and (2),... 

A tableau may be used to define certain subgroups of which are themselves direct products of smaller permutation groups the symmetrizing and antisymmetrizing operators for these subgroups lead, as we shall see, to projection operators on irreducible representations of... 

A. Projection onto the Irreducible Representations of the nth-Order Symmetric Group... 

Determination of a wave function for a system that obeys the correct permutational symmetry may be ensured by projection onto the irreducible representations of the symmetry groups to which the systems in question belong. For each subset of identical particles i, we can implement the desired permutational symmetry into the basis functions by projection onto the irreducible representation of the permutation group, for total spin 5, using the appropriate projection operator T,. The total projection operator would then be a product ... 

This requires that the eigenfunctions of the Hamiltonian are simultaneously eigenfunctions of both the Hamiltonian and the symmetric group. This may be accomplished by taking the basis functions used in the calculations, which may be called primitive basis functions, and projecting them onto the appropriate irreducible representation of the symmetric group. After this treatment, we may call the basis functions symmetry-projected basis functions. 

Projection operators for irreducible representations of the symmetric group are obtained easily from their corresponding Young tableaux. A Young tableau is created from a Young frame. A Young frame is a series of connected boxes such as... 

The correct symmetry of the system may be ensured by projection onto the irreducible representations, as described above. Thus, when taking symmetry into account, the final form of the basis function is... 

The correct permutational symmetry was implemented into the wave functions by projection onto irreducible representations of the total symmetry group for heteronuclear species and for homonuclear species, where e refers to electron exchange and H refers to nuclear exchange. The irreducible representations chosen were singlets in all cases. 

Hyperspherical harmonics are now explicitly considered as expansion basis sets for atomic and molecular orbitals. In this treatment the key role is played by a generalization of the famous Fock projection [5] for hydrogen atom in momentum space, leading to the connection between hydrogenic orbitals and four-dimensional harmonics, as we have seen in the previous section. It is well known that the hyperspherical harmonics are a basis for the irreducible representations of the rotational group on the four-dimensional hypersphere from this viewpoint hydrogenoid orbitals can be looked at as representations of the four-dimensional hyperspherical symmetry [14]. 

Here, Cys is the Cl vector in the basis of VB structures, projected such that it transforms according to the irreducible representation phi. Because even the standard CASVB approach involves an expansion of Fvb in terms of structures formed from orthogonal molecular orbitals (the transformation given in Eq. (41)), this implementation is completely straightforward. 

We shall introduce the technique of projection operators to determine the appropriate expansion coefficients for symmetry-adapted molecular orbitals. Projection by operators is a generalization of the resolution of an ordinary 3-vector into x, y and z components. The result of applying symmetry projection operators to a function is the expression of this function as a sum of components each of which transforms according to an irreducible representation of the appropriate symmetry group. 

Projection operators are a technique for constructing linear combinations of basis functions that transform according to irreducible representations of a group. Projection operators can be used to form molecular orbitals from a basis set of atomic orbitals, or to form normal modes of vibration from a basis of displacement vectors. With projection operators we can revisit a number of topics considered previously but which can now be treated in a uniform way. 

Separating the even and odd components of the function F, by means of the projection operators F- and F produces functions that transform according to irreducible representations Ag and A of the group Ci, which consists of symmetry elements E and i. An analogous technique could be used to con-stmct functions symmetric and antisymmetric with respect to a mirror plane or a dyad. 

It is no accident that the coefficients of the operators E and i in the projection operators are the same as the character vectors in the table of irreducible representations of Q ... 

From the information on the right side of the C3V character table, translations of all four atoms in the z, x and y directions transform as Ai(z) and E(x,y), respectively, whereas rotations about the z(Rz), x(Rx), and y(Ry) axes transform as A2 and E. Hence, of the twelve motions, three translations have Ai and E symmetry and three rotations have A2 and E symmetry. This leaves six vibrations, of which two have Ai symmetry, none have A2 symmetry, and two (pairs) have E symmetry. We could obtain symmetry-adapted vibrational and rotational bases by allowing symmetry projection operators of the irreducible representation symmetries to operate on various elementary cartesian (x,y,z) atomic displacement vectors. Both Cotton and Wilson, Decius and Cross show in detail how this is accomplished. 

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