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Chapter 16: Problem 16
Verify the result (16-112).
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Consider the symmetry group of a regular tetrahedron. (a) What is the order of this group? (b) Decompose it into classes. (c) Construct its character table.
Consider the following two elements of the symmetric group \(S_{5}\) : $$ \begin{aligned} &g_{1}=[54123]=(135)(24) \\ &g_{2}=[2 \mathrm{I} 534]=(12)(345) \end{aligned} $$ Find a third element \(g\) of this group such that $$ g^{-1} g_{1} g=g_{2} $$
If \(\sigma_{x}, \sigma_{y}\), and \(\sigma_{z}\) denote the Pauli spin matrices (16-109), show that $$ \left\\{\sigma_{i}, \sigma_{j}\right\\}=2 \delta_{i j} $$ where \(\left\\{\sigma_{i}, \sigma_{j}\right\\}=\sigma_{i} \sigma_{j}+\sigma_{j} \sigma_{i}\) is the anticommutator of \(\sigma_{i}\) and \(\sigma_{j}\).
(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.
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.
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