Chapter 16: Problem 16
Verify the result (16-112).
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Considet a group \(G\) and a nonfaithful representation \(D\). Let \(G^{\prime}\) be the set of all group elements which are respresented by the unit matrix. (a) Show that \(G^{\prime}\) is a subgroup of \(G\). (b) Show that \(G^{\prime}\) is in fact a normal subgroup of \(G\). (c) If \(h^{\prime}\) is the order of \(G^{\prime}\), show that every matrix in \(D\) is the representative of \(h\) ' distinct group elements.
Consider the group of all displacements in three-dimensional space: $$ x^{\prime}=x+a \quad y^{\prime}=y+b \quad z^{\prime}=z+c $$ (a) How many parameters does this group have? (b) Construct the infinitesimal operators (in differential form). (c) Show that all the infinitesimal operators commute with each other.
Denote the elements in one class of a group by \(A_{1}, A_{2}, \ldots, A_{n_{A}}\), those in another class by \(B_{1}, B_{2}, \ldots, B_{n_{D}}\), and so on. Let \(A\) denote the element \(\sum_{i=1}^{n_{1}} A_{l}\) of the group algebra, and similarly for \(B, \ldots\). (a) Consider a particular \(n\)-dimensional irreducible representation \(D\) of the group. Show that \(\sum_{i-1}^{m A}, D\left(A_{i}\right)\), which may be abbreviated \(D(A)\), equals a constant multiple of the \(n\)-dimensional unit matrix. Evaluate the constant, in terms of \(n, n_{A}\), and \(\chi(A)\), the character of the class \(A\) in the representation \(D\). (b) If the two elements \(A\) and \(B\) of the group algebra are multiplied together, show that the product consists of complete classes; that is, we may write $$ A B=\sum_{c} s_{c} C $$ where the \(s_{c}\) are nonnegative integers. Hint : First show that \(g^{-1} A B g=A B\) for all \(g\) in the group. (c) Show that \(\left.n_{\mathcal{A}} \chi(A) n_{B}\right)(B)=n \sum_{c} s_{\mathrm{c}} n_{c} \chi(C)\). Such algebraic relations among characters are often useful in evaluating characters.
(a) Show that \(\left[X_{i}, X_{j}\right]=-\left[X_{j}, X_{i}\right]\) implies \(c_{i j}^{k}=-c_{j t}^{k}\). (b) Show that the Jacobi identity $$ \left[\left[X_{i}, X_{j}\right], X_{k}\right]+\left[\left[X_{l}, X_{k}\right], X_{i}\right]+\left[\left[X_{k}, X_{i}\right], X_{j}\right]=0 $$ implies \(\begin{aligned} c_{i j}^{l} c_{i k}^{m}+c_{j k}^{l} c_{L i}^{m}+c_{k i}^{l} c_{l j}^{m}=0 & \text { (summation on repeated indices is } \\ & \text { implied) } \end{aligned}\) [Conditions \((a)\) and \((b)\) turn out to be the only conditions on the structure constants; any set of real numbers \(c_{i j}^{i}\) obeying these two conditions defines a Lie algebra.]
(a) Let \(M_{1}, M_{2}, \ldots, M_{n}\) be an arbitrary set of matrices, and let a Hermitian matrix \(K\) have the property $$ M_{i}^{+} K M_{1}=K $$ for all \(i\). Then, if all the eigenvalues of \(K\) are positive, show that a Hermitian matrix \(H\), with \(H^{2}=K\), exists such that \(H M_{i} H^{-1}\) is unitary for all \(i\). (b) If \(D(g)\) is a representation of a finite group of order \(n\), show that \(K=\sum_{i=1}^{n} D^{+}\left(g_{i}\right) D\left(g_{i}\right)\) has the properties (1) \(K=K^{+}\) (2) All eigenvalues of \(K\) are positive (3) \(D^{+}\left(g_{k}\right) K D\left(g_{k}\right)=K\) for all \(k\) and hence the representation \(I(g)\) can be made unitary by a similarity transformation.
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