Characters of Representations

Let me see four times five is twelve, and four times six is thirteen, and four times seven is — oh dear I shall never get to twenty at that rate  

In this section we define characters. Associated to each finite-dimensional representation (G, V, p) is a complex-valued function on the group G, called the character of the representation Recall the trace of an operator (Definition 2,8) the sum of the diagonal elements of the corresponding matrix, expressed in any basis. 

Definition 4.13 Suppose G, V, p) is a representation with finite-dimension-al vector space V. Define the character /p G C of the representation /p(,g) = Trp(g), 

For example, consider the representation of (S[/(2), C, p), where p is given by matrix multiplication  

We would like to find the character of each representation of 5(7(2) on homogeneous polynomials in two variables, introduced in Section 4.6. For each nonnegative integer n it suffices to find the diagonal entries of the matrix form of the transformation R g) in the familiar basis. We calculated some of 

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We are most interested in integrating products of characters of representations. In this case, we can use the Spectral Theorem (Proposition 4.4) to simplify the expression of the integral. The proposition implies that for any function f invariant under conjugation, we have... 

Since the Bravais cell contains lO atoms, it has 3x10 — 3 = 27 normal vibrations, excluding three translational motions of the cell as a whole.l These 27 vibrations can be classified into various symmetry species of the factor group Djd, using a procedure similar to that described in Sec. 1-7 for internal vibrations. First, we calculate the characters of representations corresponding to the entire freedom possessed by the Bravais primitive cell [a r( )] translational motions of the whole cell [ (T)], translatory lattice modes [ f(T )],... 

JV)= p)[ (i + 2cos )), character of representation for entire freedom possessed by the primitive cell. 

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