Isomorphic representation group

Proposition 4.6 Suppose (G, V, p) and (G, W, p) are isomorphic representations of the group G. Then either both V and W are infinite-dimensional, or both are finite dimensional and the dimension of V is equal to the dimension ofW. 

Exercise 4.40 (Used in Proposition 6.12) Suppose p and p are isomorphic representations of a group G. Show that their characters are equal. 

Hence, we see that the isomorphic representation of the group Hn becomes abelian. According to eq.(22), the classical observable B, that is a function of the phase-space variables (q,p), can be written as... 

The transformation matrix is orthogonal of order 2. With every element T() of the group can be associated a 2 x 2 orthogonal matrix with determinant +1 and the correspondence is one-to-one. The set of all orthogonal matrices of order 2 having determinant +1 is a group isomorphic to 0(2) and therefore provides a two-dimensional representation for it. The matrix group is also denoted by the symbol 0(2). 

The group (E, J) has only two one-dimensional irreducible representations. The representations of 0/(3) can therefore be obtained from those of 0(3) as direct products. The group 0/(3) is called the three-dimensional rotation-inversion group. It is isomorphic with the crystallographic space group Pi. 

The examples used above to illustrate the features of the software were kept deliberately simple. The utility of the symbolic software becomes appreciated when larger problems are attacked. For example, the direct product of S3 (order 6) and S4 (isomorphic to the tetrahedral point group) is of order 144, and has 15 classes and representations. The list of classes and the character table each require nearly a full page of lineprinter printout. When asked for, the correlation tables and decomposition of products of representations are evaluated and displayed on the screen within one or two seconds. Table VII shows the results of decomposing the products of two pairs of representations in this product group. 

In particular, a group of numbers isomorphic to a symmetry group is an example of a representation of the symmetry group. Group representations are of the utmost importance in chemistry because they make it possible to achieve the effects of geometrical reasoning by means of calculations with the numerical representations. 

We are now ready for the main conclusion of this chapter If all elements of a symmetry group are represented by orthogonal matrices in a consistent coordinate system, the matrices will form a group under the operation of matrix multiplication that is isomorphic to the symmetry group. The set of matrices is said to be a representation of the group. 

If each group element corresponds to a different matrix, the representation is said to ht faithful. A faithful representation is a matrix group that is isomorphic to the group being represented. 

Just as the same group can arise in different guises, two different-looking representations can be essenhally the same. Hence it is useful to dehne isomorphisms of representahons. Homomorphisms of representations play an important role in the critical technical tools developed in Chapter 6. We will also use them in the proof of Proposition 11.1. 

It is useful to have a shorthand for the statement that (G, V, p) is isomorphic to (G, W, pf One can write p = p. A notation common in the literature is y = W. This last shorthand puts the burden on the reader to determine from context what the group and the representations are. 

This character is the same as the character of the representation on by matrix multiplication in fact, these two representations are isomorphic, as the reader may show in Exercise 4.36. This is an example of the general phenomenon that will help us to classify representations finite-dimensional representations are isomorphic if and only if their characters are equal. See Proposition 6.12. Note that while a representation is a relatively complicated object, a character is simply a function from a group to the complex numbers it is remarkable that so much information about the complicated object is encapsulated in the simpler object. 

Proposition 6.11 Suppose (G, V, p) is a finite-dimensional representation ofi a compact group G. Then there are a finite number of distinct (i.e., not isomorphic) irreducible representations (G, Wj, pj) such that... 

By analogy with our notation for group representations, we denote a representation by a triple (g, V, p or, when the rest is clear from context, simply by V or p. As for groups, we define homomorphisms and isomorphisms of representations. 

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). 

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