Tensor Products of Representations

Next we define tensor product representations. The reader may wish to recall the definition of the tensor product of two vector spaces, given in Definition 2.14. 

Note that although not every element of V V is a one-term product of the 

The character / of the representation is the trace of the diagonal 6x6 matrix whose entries we have just calculated  

Using the result of Exercise 4.39, it is easy to calculate that / = /i/2- This is a special case of the general truth that the character of a tensor product is the product of the characters of the factors. See Proposition 5.8. 

Note also that y = yi + From Proposition 5.7 we know that the rep- 

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Not only are linear transformations necessary for the very definition of a representation in Chapter 6, but they are useful in calculating dimensions of vector spaces — see Proposition 2.5. Linear transformations are at the heart of homomorphisms of representations and many other constructions. We will often appeal to the propositions in this section as we construct linear transformations. For example, we will use Proposition 2.4 in Section 5.3 to define the tensor product of representations. 

Another useful way to construct a vector space from other vector spaces is to take what mathematicians call a tensor product and physicists call a direct product. We will need to consider tensor products of representations in Section 5.3. In this section we will define and discuss tensor products of vector spaces. 

Tensor products of representations arise naturally in physics. To obtain the space of states of two particles, take the tensor product of the two spaces of states. Thus the state space for two mobile particles in R is 0... 

We will use Cartesian sums and tensor products to build and decompose representations in Chapters 5 and 7. Tensor products are useful in combining different aspects of one particle. For instance, when we consider both the mobile and the spin properties of an electron (in Section 11.4) the state space is the tensor product of the mobile state space defined in Chapter 3)... 

The direct sum and tensor product of linear representations are again representations, so the corresponding constructions necessarily work for comodules. If U and V are comodules, for instance, then... 

The correlation function can be written in a basis set Ra) = R) a) chosen as the tensor product of the coordinate representation for the nuclear degrees of freedom, and a nuclear coordinate independent electronic basis. In this chapter we shall refer to this electronic representation as the diabatic basis. (We refer the interested reader to reference [40] for the development of this approach with a more general electronic representation). By inserting resolutions of the identity in this basis, one obtains... 

Weyl answered the first point with the insight that all irreducible representations of the special linear group can be made as invariant subspaces of tensor powers of the underlying standard representation. They were conceived as operations of the linear group transformations with a determinant on a geometrical coordinate space. Any representation of the linear group can be characterized with a tensor product of the coordinate space by a symmetry property. 

The zeroth order degeneracy of K and T-states is lifted in the spectral representation of the fully interacting Green s function. The exact eigenstates decouple to yield A -particle states in the second component and tensor products of W — 1 and N -L 1-particle states in the third component. We will come back to discuss the meaning of the degenerate states A >) in connection with the lowest order of the static self energy S oo) in Sec. V. 

Using the postulate IV, it is possible to construct the Hilbert space for systems containing two or more qubits. For a two-qubit system, the dimension of the Hilbert space is 4 X 4, since it is composed by vectors (kets) and matrices (operators), calculated using the tensor product of each vector and matrix for the individual qubit, as may be seen on Equations (3.4.10) and (3.4.11), where both representations, kets and vectors, are shown ... 

With the representations of Section A.8 one can form tensor products. Tensor products are usually denoted by the symbol ,... 

In this picture, the correspondence between irreducible representations of F (except the trivial representation) and irreducible components of the exceptional set becomes concrete. It is realized by the tautological bundles V s. In [66, 5.8], we have shown the correspondence respects the multiplicative structures, one given by the tensor product and one given by the cup product. In fact, using (4.11), we can show that two matrices... 

We denote the tensor of such elements as Dp, which is the tensor representation of the kernel Dp in a basis of p-electron direct products of the spin orbitals 4>j) [46]. The convention introduced in Eq. (8), that the number of indices implicitly specifies the tensor rank, is followed wherever tensors are used in this chapter. 

Proposition 5.8 Suppose p and p are two finite-dimensional representations of the same group G. Let / denote the character of p and let / denote the character of p. Then the character of the tensor product representation p p is the function / f G C. 

If both factors are unitary representations, then so is the tensor product. If both V and V have complex scalar products defined on them, then there is a natural complex scalar product on the tensor product V 0 V of vector spaces. Specifically, we define... 

The results of this section, even with their limitations, are the punch line of our story, the particularly beautiful goal promised in the preface. Now is a perfect time for the reader to take a few moments to reflect on the journey. We have studied a significant amount of mathematics, including approximations in vector spaces of functions, representations, invariance, isomorphism, irreducibility and tensor products. We have used some big theorems, such as the Stone-Weierstrass Theorem, Fubini s Theorem and the Spectral Theorem. Was it worth it And, putting aside any aesthetic pleasure the reader may have experienced, was it worth it from the experimental point of view In other words, are the predictions of this section worth the effort of building the mathematical machinery ... 

In the second-quantization representation the atomic interaction operators are given by relations (13.22) and (13.23), which do not include the operators themselves in coordinate representations, but rather their one-electron and two-electron matrix elements. Therefore, in terms of irreducible tensors in orbital and spin spaces, we must expand the products of creation and annihilation operators that enter (13.22) and (13.23). In this approach, the tensorial properties of one-electron wave functions are translated to second-quantization operators. 

As an illustration of dynamical supersymmetry with canonical decomposition we construct a supermultiplet starting from the even-even nucleus 194Pt. In this region the proton shell is dominated by j = 3/2 and the neutron shell by j = 1/2,3/2,5/2. Hence n = 4 and n =12. The relevant representation of the appropriate supergroup U(6/16) is the one with Jf= 7. Various nuclei are placed in the tensor product representations as follows ... 

Inserting resolutions of the identity written in terms of tensor product states RjAajA) (with 0 < j < [n — 1)) in the coordinate and diabatic state representation, matrix elements of the time dependent density operator are conveniently written as... 

Up to this point, our main concern was to reformulate the results of the LD ligand influence theory in the DMM form. Its main content was the symmetry-based analysis of the possible interplay between two types of perturbation substitution and deformation, controlled by the selection rules incorporated in the polarization propagator of the CLS. The mechanism of this interplay can be simply formulated as follows substitution produces perturbations of different symmetries which are supposed to induce transition densities of the same symmetries. In the frontier orbital approximation, only those densities among all possible ones can actually appear, which have the symmetry which enters into decomposition of the tensor product TH TL to the irreducible representations. These survived transition densities then induce the geometry deformations of the same symmetry. 

Thus it may produce the transitional densities of the alg, egc, and tluz symmetries. At this point selection rules pertinent to the frontier orbitals approximation enter for the 12-electron complexes the symmetries of the frontier orbitals are Th = eg and Tl = ai3, the tensor product Th <8> TL = eg aig = eg contains only the irreducible representation eg so that the selection rules allow only the density component of the egc symmetry to appear. In its turn this density induces additional deformation of the same symmetry. That means that in the frontier orbitals approximation, only the elastic constant for the vibration modes of the symmetry eg is renormalized. This result is to be understood in terms of individual nuclear shifts of the ligands in the trans- and cis-positions relative to the apical one. They, respectively, are ... 

By contrast, for the 14-electron complexes (nontransition nonmetals) the symmetries of the frontier orbitals are I n = aig and rL = l u and the tensor product Th rL = a g 11 = 11 so that only the transition density corresponding to the representation t survive. Analogous moves allow us to conclude that the off-diagonal elastic constant for stretching the /.ran.s-bonds has the form ... 

Theorem. Let k be a field, G a closed subgroup of GL . Every finitedimensional representation of G can be constructed from its original representation on fc" by the processes of forming tensor products, direct sums, subrepresentations, quotients, and duals. 

S]). The direct piezoelectric effect is the production of electric displacement by the application of a mechanical stress the converse piezoelectric effect results in the production of a strain when an electric field is applied to a piezoelectric crystal. The relation between stress and strain, expressed by Equation 2.7, is indicated by the term Elasticity. Numbers in square brackets show the ranks of the crystal property tensors the piezoelectric coefficients are 3rd-rank tensors, and the elastic stiffnesses are 4th-rank tensors. Numbers in parentheses identify Ist-rank tensors (vectors, such as electric field and electric displacement), and 2nd-rai tensors (stress and strain). Note that one could expand this representation to include thermal variables (see [5]) and magnetic variables. 

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