## Homework 14

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#### Q14.1 \(\quad\) number of irreducible representations

Group \( C_{nv}=\{e,C_n,C_n^2,\cdots,C_n^{n-1},\sigma,C_n\sigma,\cdots, C_{n}^{n-1}\sigma\} \) denotes the symmetry group of a horizontal plane, which is invariant under rotation about the z-axis by \(2\pi/n\), and also invariant under reflection with respect to a perpendicular plane.Please figure out the multiplication table of this discrete group, find the number of classes. How many irreducible representations does \(C_{10v}\) have ? Please think about the general case. Note, even or odd \(n\) results in different formulae.

#### Q14.2 \(\quad\) Character table of \(D_2\) (or \(C_{2v}\))

Group \( D_{2}=\{ e, C^{1/2}, \sigma, C^{1/2}\sigma \}\) is an Abelian group, so it has 4 different irreducible representations. The incomplete character table of \(D_2\) is listed below \[ \begin{bmatrix} & e & C^{1/2} & \sigma & C^{1/2}\sigma \\ A_1 & 1 & 1 & 1 & 1 \\ A_2 & 1 & 1 & -1 & -1 \\ B_1 & & & & \\ B_2 & & & & * \\ \end{bmatrix} \] Here \(A_1\) and \(A_2\) are both symmetric irreducible representations whose basis functions are symmetric under rotation \(C^{1/2}\), but the basis function of \(A_2\) is antisymmetric under reflection \(\sigma\). \(B_1\) and \(B_2\) are both antisymmetric irreducible representation. The difference is that the basis function of \(B_2\) is symmetric under reflection \(\sigma\). What is the number at position (*) ? Actually, there are 2 discrete groups of order 4, and both of them are Abelian. The other is \(C_4\). If you like, think about whether these two groups are isomorphic to each other or not.#### Q14.3 \(\quad\) Invariant basis of \(C_{4v}\)

Each irreducible representation of a group is associated with an invariant subspace. As stated in Chaosbook, projection operator is the right tool to find the invariant subspaces of a group. For group \(C_{4v} = \{e, C^{1/4}, C^{2/4}, C^{3/4}, \sigma, \sigma C^{1/4}, \sigma C^{2/4}, \sigma C^{3/4} \} \), first figure out its character table, and then use projection operator to figure out all the invariant subspaces.For the following candidates, which one is

**NOT**an invariant subspace of \(C_{4v}\)? \begin{align*} \rho_0(\hat{x}) &=& \frac{1}{8}\left[ \rho(\hat{x}) ~+~ \rho(C^{2/4} \hat{x}) ~+~ \rho(C^{1/4} \hat{x}) ~+~ \rho(C^{3/4} \hat{x}) ~+~ \rho(C^{2/4} \sigma \hat{x}) ~+~ \rho(\sigma \hat{x}) ~+~ \rho(\sigma C^{1/4} \hat{x}) ~+~ \rho(\sigma C^{3/4} \hat{x}) \right] \\ \rho_1(\hat{x}) &=& \frac{1}{8}\left[ \rho(\hat{x}) ~+~ \rho(C^{2/4} \hat{x}) ~-~ \rho(C^{1/4} \hat{x}) ~-~ \rho(C^{3/4} \hat{x}) ~-~ \rho(\sigma C^{2/4} \hat{x}) ~-~ \rho(\sigma \hat{x}) ~+~ \rho(\sigma C^{1/4} \hat{x}) ~+~ \rho(\sigma C^{3/4} \hat{x}) \right] \\ \rho_2(\hat{x}) &=& \frac{1}{8}\left[ \rho(\hat{x}) ~+~ \rho(C^{2/4} \hat{x}) ~-~ \rho(C^{1/4} \hat{x}) ~-~ \rho(C^{3/4} \hat{x}) ~+~ \rho(C^{2/4} \sigma \hat{x}) ~+~ \rho(\sigma \hat{x}) ~-~ \rho(\sigma C^{1/4} \hat{x}) ~-~ \rho(\sigma C^{3/4} \hat{x}) \right] \\ \rho_3(\hat{x}) &=& \frac{1}{8}\left[ \rho(\hat{x}) ~+~ \rho(C^{2/4} \hat{x}) ~+~ \rho(C^{1/4} \hat{x}) ~+~ \rho(C^{3/4} \hat{x}) ~-~ \rho(C^{2/4} \sigma \hat{x}) ~-~ \rho(\sigma \hat{x}) ~-~ \rho(\sigma C^{1/4} \hat{x}) ~-~ \rho(\sigma C^{3/4} \hat{x}) \right] \\ \rho_4(\hat{x}) &=& \frac{1}{8}\left[ \rho(\hat{x}) ~-~ \rho(C^{2/4} \hat{x}) ~+~ \rho(C^{1/4} \hat{x}) ~+~ \rho(C^{3/4} \hat{x}) ~-~ \rho(C^{2/4} \sigma \hat{x}) ~+~ \rho(\sigma \hat{x}) ~-~ \rho(\sigma C^{1/4} \hat{x}) ~-~ \rho(\sigma C^{3/4} \hat{x}) \right] \end{align*}

\( \rho_0 \) \( \rho_1 \) \( \rho_2 \) \( \rho_3 \) \( \rho_4 \)

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