(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.