Determinants as representations

We might note, in passing that, since det(A ) — det(A)det(i ) (see Appendix A.4-6), the determinants of any set of matrices which form a representation will themselves act as a one-dimensional representation. That is, if T = SR and D(T) = D(S)D(R) then 

In this chapter we have shown that there are very many different sets of matrices which behave like the symmetry operations of a given point group. We have constructed these so-called representations by considering the action of the symmetry operations on a position vector or on any number of base vectors. Alternatively, we have found that we can find transformation operators Om which are homomorphic with the symmetry operations and that from these we can construct 

Our next task will be to try and organize, reduce and classify the plethora of representations which we can now create. We will try to eliminate from discussion those which are, in a certain sense, equivalent and those which can, in a certain sense, be broken down into simpler (lower order) representations. 

Where we have now got to and where we are going has already been summarized in Fig. 5-1.1. 

Proof that tha matrices constructed in aqn 5-4.2 (or aqn 5-4.4) form a raprasantation of tha point----------- 

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