Isomorphism of representations

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

Next we introduce isomorphisms of representations. As usual, isomor-... 

Isomorphisms of unitary representations ought to preserve the unitary structure. When they do, they are called unitary isomorphisms of representations. 

Now suppose that the representations Vi and V2 are indeed isomorphic. Let T and f denote isomorphisms (of representations) from V to 72- It suffices to show that T must be a scalar multiple of T. Consider the linear transformation T o V2 V2- By Exercise 4.19, this linear transformation is an isomorphism of representations. Hence by Proposition 6.3, there must be a complex number A such that T o 7 = kl, and hence t = XT. Note that because 7 is an isomorphism, A 0. ... 

Finally, we must show that if k > k, then VF is not isomorphic to a subrepresentation of V. We prove the contrapositive. Suppose that k e N and there is an isomorphism of representations from Wk to a subrepresentation of V. Then... 

From Proposition 5.1 we know that y is an invariant subspace. Since the natural representation of S<9(3) on L (W ) is unitary. Proposition 5.4 implies that is a homomorphism of representations. Since V and y are irreducible, it follows from Schur s Lemma and the nontriviality of n [V] that fit gives an isomorphism of representations from V to y. ... 

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. 

We can use the basis 5 to define a linear transformation T V P . This transformation will turn out to be an isomorphism of representations. Define T V P" by... 

It remains to check that T is an isomorphism of representations, Eor the remaining condition of Definition 8.5, it suffices to check that for any k = 0,. .., we have... 

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. 

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. 

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... 

Now W is a finite-dimensional, unitary, irreducible representation, so by Proposition 6.16 there must be a nonnegative even integer h and an isomorphism T P W of representations. Because T is an isomorphism, the fist of weights for P must be the same as the fist of weights for W. Hence... 

Then we say that T is a homomorphism of (Lie algebra) representations. If in addition T is injective and surjective then we say that T is an isomorphism of (Lie algebra) representations and that p is isomorphic fo p. 

Find the representations of the space group 227 (Fd3m or O],) at the surface point B(1/2 + /3, ( I 3, Vi + a), point group Cs= E ay. [Hints-. Use the method of induced representations. Look for an isomorphism of Cs with a cyclic point group of low order. The multiplication table of Cs will be helpful.]... 

Proposition 1.3. — Let f T —> T2 be a reasonable morphism of sites. Assume in addition that T2 has products (but not fiber products ) and that the functor commutes with them. Let further M ie a sheaf of simplicial monoids on T2 such that all the terms M, considered as sheaves of sets are direct sums of representable sheaves. Then there is a natural (in Mj isomorphism in. i(Ti) of the form... 

Proof. — Using Lemma 1.1 and Proposition 1.57(2) we may assume that each term of M is the sheaf of monoids freely generated by a direct sum of representable sheaves of sets. Since f commutes with products of representable sheaves / (M) is again a monoid with the same property. By Lemma 1.2 it remains to define an isomorphism I (M ) ( (M))". We clearly have (/ (M))" =/ (M ) which means by... 

Figure B3.3.11. The classical ring polymer isomorphism, forA = 2 atoms, using/ = 5 beads. The wavy lines represent quantum spring bonds between different imaginary-time representations of the same atom. The dashed lines represent real pair-potential interactions, each diminished by a factor P, between the atoms, linking corresponding imaginary times.

Structure searching is the chemical equivalent of graph isomorphism, that is, the matching of one graph against another to determine whether they are identical. This can be carried out very rapidly if a unique structure representation is available, because a character-by-character match will then suffice to compare two structures for identity. However, connection tables are not necessarily unique, because very many different tables can be created for the same molecule depending upon the way in which the atoms in the molecule are numbered. Specifically, for a molecule containing N atoms, there are N ... 

---