Homomorphism representations

Because of the way in which block-diagonal matrices multiply, the 2 by 2 bloeks in the matrices in Eq. (9.76) if taken alone form another representation of the Csy group. When a reducible representation is written with its matrices in block-diagonal form, the block submatrices form irreducible representations, and the reducible representation is said to be the direct sum of the irreducible representations. Both the representations obtained from the submatrices are irreducible. The 1 by 1 blocks form an unfaithful or homomorphic representation, in which every operator is represented by the 1 by 1 identity matrix. This one-dimensional representation is called the totally symmetric representation. In this particular case we could not get three one-dimensional representations, because the Cs(z) operator mixes the x and y coordinates of a particle, preventing the matrices from being diagonal. 

Note. These representations do not follow the recommendations for choice of main chain given in 2-Carb-37.3. Such deviations are common in depicting series of naturally occurring oligosaccharides where it is desirable to show homomorphic relationships. 

In an n-dimensional space L, the linear operators of the representation can be described by their matrix representatives. This procedure produces a homomorphic mapping of the group G on a group of n x n matrices D(G), i.e., a matrix representation of the group G. From equations (6) it follows that the matrices are non-singular, and that... 

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. 

There is an important surjective group homomorphism from 5 U (2) to 50(3). We will find the homomorphism useful in Section 6.6 for deriving the list of irreducible representations of 50(3) from the list of irreducible representations of 50(2). There is no a priori reason to expect such a homomorphism between two arbitrary groups, so the fact that 50(2) and 50(3) are related in this way is quite special. Here is the definition of 4> ... 

In this section we define representations and give examples. We also define homomorphisms and isomorphisms of representations, as well as unitary representations and isomorphisms. 

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. 

If r is a homomorphism of representahons, then T is said to intertwine the two representations. Because the condihon for T to be a homomorphism is linear in T, it follows that the set of homomorphisms of representahons from V to W forms a vector space. 

Note that every unitary homomorphism T of representations is injective if V 7 0 e V, then... 

Any physical representation p must also be a Lie group homomorphism ... 

Exercise 4.15 (Used in Proposition 5.1) Show that the Laplacian in three variables is invariant under rotation. In other words, consider the natural representation p ofSO 3) on twice-dijferentiable functions of three variables and show that for any g e SO (3) we have p(g)oV2 = op g). To put it yet another way, show that the Laplacian is a homomorphism of representations. 

Proposition 5.4 Suppose W is an invariant subspace for a unitary representation (G, V, p). Suppose that there is an orthogonal projection flyv V V onto a subspace VP. Then FI w is a homomorphism of representations. 

Recall the projection onto the k-th summand from Definition 2.12. This projection is a homomorphism of representations. 

The set of all linear transformations (not necessarily homomorptiisms of representations) from a representation V to a representation W forms a vector space too. This vector space is denoted Hom(T, IT). (Here Hom refers to the fact that a linear transformation can be considered a homomorphism of vector spaces.) There is a natural representation of G on this vector space. 

In this section we have seen how representations on two spaces V and W determine a representation on the set of homomorphisms of representations from V to W. Familiarity with this kinds of categorical construction is often the key to finding simple, direct proofs of interesting results such as Proposition 6.8. 

In this section we show how to use group homomorphisms to construct a representation of one group from a representation of another group. 

Proof. First we show that (G, V, p o F) is a representation by checking the criteria given in Definition 4.7. We know by hypothesis that G is a group and V is a vector space. Because both p and 4 are group homomorphisms, it follows from Proposition 4.3 that p o group homomorphism from G to QC (V). Hence p o 4 is a representation. 

Consider for example the inclusion map i of a subgroup G of a group G, defined in Exercise 4.37. By that exercise, the inclusion map is a group homomorphism. Note that for any representation p of G, the pullback representation p o z is just the restriction of p to the subgroup G. 

It is not usually possible to push a representation forward, i.e., to use a representation on the domain of a group homomorphism to obtain a representation on the image. See Exercise 5.4. However, in certain circumstances a pushforward representation can be defined. See Figure 5.3. 

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