Appendix I — Homework 8
Exercise I.1 Let \(\theta \in \mathbb{R}\) and \(n \in \mathbb{Z}^+\). Prove the following identity: \[ e^{n \theta i} + e^{(n-2) \theta i} + \cdots + e^{-(n-2) \theta i} + e^{-n \theta i} = \frac{\sin(n+1)\theta}{\sin \theta}. \]
Exercise I.2 Consider the Hamilton quaternion algebra \[ \mathbb{H}= \left\{ a+bi+cj+dk \mid a, b, c, d \in \mathbb{R}\right\}, \] where the symbols \(i, j,\) and \(k\) obey the relations from the quaternion group \(\mathcal{Q}_{8}\). We define an involution on \(\mathbb{H}\) by \((a+bi+cj+dk)^* \coloneqq a-bi-cj-dk\) and a norm by \[ \|a+bi+cj+dk\| \coloneqq \sqrt{a^2+b^2+c^2+d^2}. \]
Consider \(\mathbb{V}= \operatorname{Span}\{i,j,k\}\), which we identify with \(\mathbb{R}^3\) via \(bi+cj+dk \mapsto \begin{psmallmatrix} b \\ c \\ d \end{psmallmatrix}\). If \(x, y \in \mathbb{V}\), show \[ xy = \underbrace{x \times y}_{\in \mathbb{V}} - \underbrace{x \cdot y.}_{\in \mathbb{R}} \]
Check that \((qp)^* = p^* q^*\), \(q^*q = \|q\|^2\), and \(\|qp\|=\|q\|\|p\|\) for all \(q, p \in \mathbb{H}\).
Hint: It helps to use \(\mathbb{H}= \mathbb{R}\oplus \mathbb{V}\).
Compute the center \(\mathbf{Z}(\mathbb{H})\).
Hint: If \(q = t+x, p = s+y\) for \(t, s \in \mathbb{R}\) and \(x,y \in \mathbb{V}\), show that \(\tfrac{1}{2} [q,p] = x \times y\).1
If \(u \in \mathbb{V}\) has \(\|u\| = 1\), show that \(u^2 = -1\).
With \(u\) as before, show that \(e^{u \theta} = \cos \theta + u \sin \theta\) for all \(\theta \in \mathbb{R}\), where we define \[ e^q \coloneqq 1 + q + \tfrac{1}{2} q^2 + \cdots + \tfrac{1}{n!} q^n + \cdots. \] Conclude that every nonzero \(q \in \mathbb{H}\) can be written as \(q = r e^{u \theta}\) for \(r \in \mathbb{R}^+\), \(\theta \in [0,\pi]\), and \(u \in \mathbb{V}\) with \(\|u\| = 1\).
Exercise I.3 Here we investigate the basic structure of the compact group \[ \operatorname{SU}(2) = \{ Q \in \operatorname{GL}_2(\mathbb{C}): Q^* Q = \mathbb{I}\text{ and } \det Q = 1 \}. \]
Show that every element \(Q \in \operatorname{SU}(2)\) is of the form \(Q = \begin{psmallmatrix} \alpha & \beta \\ - \overline{\beta} & \overline{\alpha} \end{psmallmatrix}\) for \(\alpha, \beta \in \mathbb{C}\) such that \(|\alpha|^2+|\beta|^2=1\).
Consider the set of quaternions with norm 1, a group with respect to multiplication: \[ \mathbb{S}^3 \coloneqq \left\{ \alpha+\beta j \mid \alpha, \beta \in \mathbb{C}, |\alpha|^2+|\beta|^2=1 \right\} \subseteq \mathbb{H}. \] Show that the following is a group isomorphism2: \[ \begin{aligned} \varphi: \mathbb{S}^3 & \to \operatorname{SU}(2) \\ \alpha + \beta j & \mapsto \begin{pmatrix} \alpha & \beta \\ - \overline{\beta} & \overline{\alpha} \end{pmatrix}. \end{aligned} \]
Hint: Notice that \(\alpha j = j \overline{\alpha}\) for all \(\alpha \in \mathbb{C}\).
Show that every \(Q \in \operatorname{SU}(2)\) is conjugate to \(\operatorname{Diag}(e^{\theta i},e^{-\theta i})\) for some unique \(\theta \in [0,\pi]\).
Exercise I.4 Now we move on to representation theory. By the prequel, we can use spherical coordinates to parameterize \(\operatorname{SU}(2)\): \[ (\theta, \psi, \phi) \mapsto \begin{pmatrix} \cos \theta + \sin \theta \sin \psi \sin \phi \; i & \sin \theta \cos \psi + \sin \theta \sin \psi \cos \phi \; i \\ -\sin \theta \cos \psi + \sin \theta \sin \psi \cos \phi \; i & \cos \theta - \sin \theta \sin \psi \sin \phi \; i \end{pmatrix}, \] where \(\theta, \psi \in [0,\pi]\) and \(\phi \in [0,2\pi)\); this matrix diagonalizes as in Exercise I.3(c). The Haar measure on \(\operatorname{SU}(2)\) is given by \[ f \mapsto \frac{1}{2\pi^2} \int_0^{2\pi} \int_0^\pi \int_0^\pi f(\theta,\psi,\phi) \sin^2 \theta \sin \psi \mathop{}\!\mathrm{d}{\theta} \mathop{}\!\mathrm{d}{\psi} \mathop{}\!\mathrm{d}{\phi}. \]
Simplify the Haar measure in the case when \(f\) is a class function, i.e., \(f = f(\theta)\).
We write \(V_n \coloneqq \mathbb{C}^{(n)}[x,y]\) for the vector space of homogeneous degree \(n\) polynomials in two variables, i.e., \[ \mathbb{C}^{(n)}[x,y] = \operatorname{Span}\{x^n, x^{n-1}y, \dots, xy^{n-1}, y^n\} = \operatorname{Sym}^{n}({(\mathbb{C}^2)^*}) \] for any \(n \in \mathbb{N}\). As usual, \(\operatorname{SU}(2)\) acts on \(V_n\) by precomposition via the inverse: \[ M \cdot p(x,y) = p( M^{-1} (x,y)), \quad \text{ i.e., } \quad (\begin{smallmatrix} \alpha & \beta \\ \gamma & \delta \end{smallmatrix}) \cdot p(x,y) = p(\tfrac{\delta x - \beta y}{\alpha \delta - \beta \gamma},\tfrac{\alpha y - \gamma x}{\alpha \delta - \beta \gamma}). \] Compute the character \(\chi_n\) of the action \(\rho_n: \operatorname{SU}(2) \to V_n\) for all \(n \in \mathbb{N}\).
Hint: You know nice representatives for the conjugacy classes of \(\operatorname{SU}(2)\)!
Show that the set \(\left\{ \chi_{n} \mid n \in \mathbb{N}\right\}\) is pairwise orthonormal, i.e., that the \(V_n\) are distinct irreducible representations of \(\operatorname{SU}(2)\).
Decompose \(V_2 \otimes V_3\) into irreducible representations.
Hint: A useful trigonometric identity is \(\frac{\sin 3 \theta}{\sin \theta} = 2 \cos 2 \theta + 1\).