• # A1.19

(i) State and prove Maschke's theorem for finite-dimensional representations of finite groups.

(ii) $S_{3}$ is the group of bijections on $\{1,2,3\}$. Find the irreducible representations of $S_{3}$, state their dimensions and give their character table.

Let $T_{2}$ be the set of objects $T_{2}=\left\{a_{i_{1} i_{2}}: i_{1}, i_{2}=1,2,3\right\}$. The operation of the permutation group $S_{3}$ on $T_{2}$ is defined by the operation of the elements of $S_{3}$ separately on each index $i_{1}$ and $i_{2}$. For example,

$P_{12}: a_{13} \rightarrow a_{23}, \quad P_{231}: a_{23} \rightarrow a_{31}, \quad P_{13}: a_{33} \rightarrow a_{11}$

By considering a representative operator from each conjugacy class of $S_{3}$, find the table of group characters for the representation $\mathcal{T}_{2}$ of $S_{3}$ acting on $T_{2}$. Hence, deduce the irreducible representations into which $\mathcal{T}_{2}$ decomposes.

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• # A3.15

(i) Show that the character of an $S U(2)$ transformation in the $2 l+1$ dimensional irreducible representation $d_{l}$ is given by

$\chi_{l}(\theta)=\frac{\sin [(l+1 / 2) \theta]}{\sin [\theta / 2]}$

What are the characters of irreducible $S O(3)$ representations?

(ii) The isospin representation of two-particle states of pions and nucleons is spanned by the basis $T=\left\{\left|\pi^{+} p\right\rangle,\left|\pi^{+} n\right\rangle,\left|\pi^{0} p\right\rangle,\left|\pi^{0} n\right\rangle,\left|\pi^{-} p\right\rangle,\left|\pi^{-} n\right\rangle\right\}$.

Pions form an isospin triplet with $\pi^{+}=|1,1\rangle, \pi^{0}=|1,0\rangle, \pi^{-}=|1,-1\rangle$; and nucleons form an isospin doublet with $p=|1 / 2,1 / 2\rangle, n=|1 / 2,-1 / 2\rangle$. Find the values of the isospin for the irreducible representations into which $T$ will decompose.

Using $I_{-}|j, m\rangle=\sqrt{(j-m+1)(j+m)}|j, m-1\rangle$, write the states of the basis $T$ in terms of isospin states.

Consider the transitions

$\begin{array}{ll} \pi^{+} p & \rightarrow \pi^{+} p \\ \pi^{-} p & \rightarrow \pi^{-} p \\ \pi^{-} p & \rightarrow \pi^{0} n \end{array}$

and show that their amplitudes satisfy a linear relation.

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• # A1.17

(i) Define the character $\chi$ of a representation $D$ of a finite group $G$. Show that $<\chi \mid \chi>=1$ if and only if $D$ is irreducible, where

$<\chi \mid \chi>=\frac{1}{|G|} \sum_{g \in G} \chi(g) \chi\left(g^{-1}\right)$

If $|G|=8$ and $<\chi \mid \chi>=2$, what are the possible dimensions of the representation $D ?$

(ii) State and prove Schur's first and second lemmas.

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• # A3.15

(i) Given that the character of an $S U(2)$ transformation in the $(2 l+1)$-dimensional irreducible representation $d_{l}$ is given by

$\chi_{l}(\theta)=\frac{\sin \left(l+\frac{1}{2}\right) \theta}{\sin \frac{\theta}{2}}$

show how the direct product representation $d_{l_{1}} \otimes d_{l_{2}}$ decomposes into irreducible $S U(2)$ representations.

(ii) Find the decomposition of the direct product representation $3 \otimes \overline{3}$ of $S U(3)$ into irreducible $S U(3)$ representations.

Mesons consist of one quark and one antiquark. The scalar Meson Octet consists of the following particles: $K^{\pm}\left(Y=\pm 1, I_{3}=\pm \frac{1}{2}\right), K^{0}\left(Y=1, I_{3}=-\frac{1}{2}\right), \bar{K}^{0}(Y=-1$, $\left.I_{3}=\frac{1}{2}\right), \pi^{\pm}\left(Y=0, I_{3}=\pm 1\right), \pi^{0}\left(Y=0, I_{3}=0\right)$ and $\eta\left(Y=0, I_{3}=0\right)^{2}$.

Use the direct product representation $3 \otimes \overline{3}$ of $S U(3)$ to identify the quark-type of the particles in the scalar Meson Octet. Deduce the quark-type of the $S U(3)$ singlet state $\eta^{\prime}$ contained in $3 \otimes \overline{3}$.

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• # A1.17

(i) Let $H$ be a normal subgroup of the group $G$. Let $G / H$ denote the group of cosets $\tilde{g}=g H$ for $g \in G$. If $D: G \rightarrow G L\left(\mathbb{C}^{n}\right)$ is a representation of $G$ with $D\left(h_{1}\right)=D\left(h_{2}\right)$ for all $h_{1}, h_{2} \in H$ show that $\tilde{D}(\tilde{g})=D(g)$ is well-defined and that it is a representation of $G / H$. Show further that $\tilde{D}(\tilde{g})$ is irreducible if and only if $D(g)$ is irreducible.

(ii) For a matrix $U \in S U(2)$ define the linear map $\Phi_{U}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ by $\Phi_{U}(\mathbf{x}) \cdot \boldsymbol{\sigma}=$ $U \mathbf{x} . \boldsymbol{\sigma} U^{\dagger}$ with $\boldsymbol{\sigma}=\left(\sigma_{1}, \sigma_{2}, \sigma_{3}\right)^{T}$ as the vector of the Pauli spin matrices

$\sigma_{1}=\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right), \quad \sigma_{2}=\left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right), \quad \sigma_{3}=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right)$

Show that $\left\|\Phi_{U}(\mathbf{x})\right\|=\|\mathbf{x}\|$. Because of the linearity of $\Phi_{U}$ there exists a matrix $R(U)$ such that $\Phi_{U}(\mathbf{x})=R(U) \mathbf{x}$. Given that any $S U(2)$ matrix can be written as

$U=\cos \alpha I-i \sin \alpha \mathbf{n} \cdot \boldsymbol{\sigma}$

where $\alpha \in[0, \pi]$ and $\mathbf{n}$ is a unit vector, deduce that $R(U) \in S O(3)$ for all $U \in S U(2)$. Compute $R(U) \mathbf{n}$ and $R(U) \mathbf{x}$ in the case that $\mathbf{x} \cdot \mathbf{n}=0$ and deduce that $R(U)$ is the matrix of a rotation about $\mathbf{n}$ with angle $2 \alpha$.

[Hint: $\mathbf{m} . \boldsymbol{\sigma} \mathbf{n} . \boldsymbol{\sigma}=\mathbf{m} . \mathbf{n} I+i(\mathbf{m} \times \mathbf{n}) . \boldsymbol{\sigma} .]$

Show that $R(U)$ defines a surjective homomorphism $\Theta: S U(2) \rightarrow S O(3)$ and find the kernel of $\Theta$.

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• # A3.15

(i) Let $D_{6}$ denote the symmetry group of rotations and reflections of a regular hexagon. The elements of $D_{6}$ are given by $\left\{e, c, c^{2}, c^{3}, c^{4}, c^{5}, b, b c, b c^{2}, b c^{3}, b c^{4}, b c^{5}\right\}$ with $c^{6}=b^{2}=e$ and $c b=b c^{5}$. The conjugacy classes of $D_{6}$ are $\{e\},\left\{c, c^{5}\right\},\left\{c^{2}, c^{4}\right\},\left\{c^{3}\right\},\left\{b, b c^{2}, b c^{4}\right\}$ and $\left\{b c, b c^{3}, b c^{5}\right\}$.

Show that the character table of $D_{6}$ is

\begin{tabular}{l|rrrrrr} $D_{6}$ & $e$ & $\left\{c, c^{5}\right\}$ & $\left\{c^{2}, c^{4}\right\}$ & $\left\{c^{3}\right\}$ & $\left\{b, b c^{2}, b c^{4}\right\}$ & $\left\{b c, b c^{3}, b c^{5}\right\}$ \ \hline$\chi_{1}$ & 1 & 1 & 1 & 1 & 1 & 1 \ $\chi_{2}$ & 1 & 1 & 1 & 1 & $-1$ & $-1$ \ $\chi_{3}$ & 1 & $-1$ & 1 & $-1$ & 1 & $-1$ \ $\chi_{4}$ & 1 & $-1$ & 1 & $-1$ & $-1$ & 1 \ $\chi_{5}$ & 2 & 1 & $-1$ & $-2$ & 0 & 0 \ $\chi_{6}$ & 2 & $-1$ & $-1$ & 2 & 0 & 0 \end{tabular}

(ii) Show that the character of an $S O(3)$ rotation with angle $\theta$ in the $2 l+1$ dimensional irreducible representation of $S O(3)$ is given by

$\chi_{l}(\theta)=1+2 \cos \theta+2 \cos (2 \theta)+\ldots+2 \cos ((l-1) \theta)+2 \cos (l \theta)$

For a hexagonal crystal of atoms find how the degeneracy of the D-wave orbital states $(l=2)$ in the atomic central potential is split by the crystal potential with $D_{6}$ symmetry and give the new degeneracies.

By using the fact that $D_{3}$ is isomorphic to $D_{6} /\left\{e, c^{3}\right\}$, or otherwise, find the degeneracies of eigenstates if the hexagonal symmetry is broken to the subgroup $D_{3}$ by a deformation. The introduction of a magnetic field further reduces the symmetry to $C_{3}$. What will the degeneracies of the energy eigenstates be now?

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• # A1.17

(i) Let $h: G \rightarrow G^{\prime}$ be a surjective homomorphism between two groups, $G$ and $G^{\prime}$. If $D^{\prime}: G^{\prime} \rightarrow G L\left(\mathbb{C}^{n}\right)$ is a representation of $G^{\prime}$, show that $D(g)=D^{\prime}(h(g))$ for $g \in G$ is a representation of $G$ and, if $D^{\prime}$ is irreducible, show that $D$ is also irreducible. Show further that $\widetilde{D}(\widetilde{g})=D^{\prime}(\widetilde{h}(\widetilde{g}))$ is a representation of $G / \operatorname{ker}(h)$, where $\tilde{h}(\widetilde{g})=h(g)$ for $g \in G$ and $\widetilde{g} \in G / \operatorname{ker}(h)$ (with $g \in \widetilde{g}$ ). Deduce that the characters $\chi, \widetilde{\chi}, \chi^{\prime}$ of $D, \widetilde{D}, D^{\prime}$, respectively, satisfy

$\chi(g)=\tilde{\chi}(\widetilde{g})=\chi^{\prime}(h(g))$

(ii) $D_{4}$ is the symmetry group of rotations and reflections of a square. If $c$ is a rotation by $\pi / 2$ about the centre of the square and $b$ is a reflection in one of its symmetry axes, then $D_{4}=\left\{e, c, c^{2}, c^{3}, b, b c, b c^{2}, b c^{3}\right\}$. Given that the conjugacy classes are $\{e\}\left\{c^{2}\right\},\left\{c, c^{3}\right\}$ $\left\{b, b c^{2}\right\}$ and $\left\{b c, b c^{3}\right\}$ derive the character table of $D_{4}$.

Let $H_{0}$ be the Hamiltonian of a particle moving in a central potential. The $S O(3)$ symmetry ensures that the energy eigenvalues of $H_{0}$ are the same for all the angular momentum eigenstates in a given irreducible $S O(3)$ representation. Therefore, the energy eigenvalues of $H_{0}$ are labelled $E_{n l}$ with $n \in \mathbb{N}$ and $l \in \mathbb{N}_{0}, l. Assume now that in a crystal the symmetry is reduced to a $D_{4}$ symmetry by an additional term $H_{1}$ of the total Hamiltonian, $H=H_{0}+H_{1}$. Find how the $H_{0}$ eigenstates in the $S O(3)$ irreducible representation with $l=2$ (the D-wave orbital) decompose into irreducible representations of $H$. You may assume that the character, $g(\theta)$, of a group element of $S O(3)$, in a representation labelled by $l$ is given by

$\chi\left(g_{\theta}\right)=1+2 \cos \theta+2 \cos (2 \theta)+\ldots+2 \cos (l \theta)$

where $\theta$ is a rotation angle and $l(l+1)$ is the eigenvalue of the total angular momentum, $\mathbf{L}^{2}$.

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• # A3.15

(i) The pions form an isospin triplet with $\pi^{+}=|1,1\rangle, \pi^{0}=|1,0\rangle$ and $\pi^{-}=|1,-1\rangle$, whilst the nucleons form an isospin doublet with $p=\left|\frac{1}{2}, \frac{1}{2}\right\rangle$ and $n=\left|\frac{1}{2},-\frac{1}{2}\right\rangle$. Consider the isospin representation of two-particle states spanned by the basis

$T=\left\{\left|\pi^{+} p\right\rangle,\left|\pi^{+} n\right\rangle,\left|\pi^{0} p\right\rangle,\left|\pi^{0} n\right\rangle,\left|\pi^{-} p\right\rangle,\left|\pi^{-} n\right\rangle\right\}$

State which irreducible representations are contained in this representation and explain why $\left|\pi^{+} p\right\rangle$ is an isospin eigenstate.

Using

$I_{-}|j, m\rangle=\sqrt{(j-m+1)(j+m)}|j, m-1\rangle,$

where $I_{-}$is the isospin ladder operator, write the isospin eigenstates in terms of the basis, $T$.

(ii) The Lie algebra $s u(2)$ of generators of $S U(2)$ is spanned by the operators $\left\{J_{+}, J_{-}, J_{3}\right\}$ satisfying the commutator algebra $\left[J_{+}, J_{-}\right]=2 J_{3}$ and $\left[J_{3}, J_{\pm}\right]=\pm J_{\pm}$. Let $\Psi_{j}$ be an eigenvector of $J_{3}: J_{3}\left(\Psi_{j}\right)=j \Psi_{j}$ such that $J_{+} \Psi_{j}=0$. The vector space $V_{j}=\operatorname{span}\left\{J_{-}^{n} \Psi_{j}: n \in \mathbb{N}_{0}\right\}$ together with the action of an arbitrary su(2) operator $A$ on $V_{j}$ defined by

$J_{-}\left(J_{-}^{n} \Psi_{j}\right)=J_{-}^{n+1} \Psi_{j}, \quad A\left(J_{-}^{n} \Psi_{j}\right)=\left[A, J_{-}\right]\left(J_{-}^{n-1} \Psi_{j}\right)+J_{-}\left(A\left(J_{-}^{n-1} \Psi_{j}\right)\right)$

forms a representation (not necessarily reducible) of $s u(2)$. Show that if $J_{-}^{n} \Psi_{j}$ is nontrivial then it is an eigenvector of $J_{3}$ and find its eigenvalue. Given that $\left[J_{+}, J_{-}^{n}\right]=$ $\alpha_{n} J_{-}^{n-1} J_{3}+\beta_{n} J_{-}^{n-1}$ show that $\alpha_{n}$ and $\beta_{n}$ satisfy

$\alpha_{n}=\alpha_{n-1}+2, \quad \beta_{n}=\beta_{n-1}-\alpha_{n-1}$

By solving these equations evaluate $\left[J_{+}, J_{-}^{n}\right]$. Show that $J_{+} J_{-}^{2 j+1} \Psi_{j}=0$. Hence show that $J_{-}^{2 j+1} \Psi_{j}$ is contained in a proper sub-representation of $V_{j}$.

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