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  "body": "\n## Example\n\nLet $G = \\mathrm{SU}(2)$ be the [[group|E5P8AG6CK7]] of complex $2 \\times 2$ unitary matrices of determinant $1$.\nFor each integer $n \\ge 0$, let $V_n$ be the space of [[homogeneous polynomials|2C3PAQ7H50]] of degree $n$ in two complex variables $z = (z_1, z_2)$, of dimension $n + 1$.\nPut $(\\pi_n(g) P)(z) = P(g^{-1} z)$ for $g \\in G$ and $P \\in V_n$.\nThis is a continuous homomorphism $G \\to \\mathrm{GL}(V_n)$, so it is a [[representation|7JA7WH92K0]] of $G$.\n\nThe classification: each $V_n$ is [[irreducible|FX3NEFYFFN]], the $V_n$ are pairwise non-isomorphic, and each irreducible finite-dimensional complex representation of $G$ is isomorphic to exactly one $V_n$.\n\nCheck of the character on $t_\\theta = \\operatorname{diag}(e^{i\\theta}, e^{-i\\theta})$, with $\\theta \\notin \\pi\\Z$.\nThe monomial $z_1^a z_2^b$ with $a + b = n$ is an eigenvector with eigenvalue $e^{i(b - a)\\theta}$, so\n$$\\tr \\pi_n(t_\\theta) = \\sum_{j=0}^{n} e^{i(n - 2j)\\theta} = \\frac{\\sin\\big((n+1)\\theta\\big)}{\\sin \\theta}.$$\nFor $n = 1$ this gives $2 \\cos \\theta$, the trace of $t_\\theta$ itself.\nSince each element of $G$ is conjugate to some $t_\\theta$, these characters determine the representations.\n\n## Remarks\n\nTensor products decompose by the Clebsch–Gordan rule: $V_m \\otimes V_n \\cong V_{m+n} \\oplus V_{m+n-2} \\oplus \\cdots \\oplus V_{\\abs{m-n}}$.\nThe element $-I$ acts on $V_n$ by $(-1)^n$, so $V_n$ is a representation of $\\mathrm{SO}(3) = \\mathrm{SU}(2)/\\set{\\pm I}$ exactly when $n$ is even.\nIn physics $V_n$ is the spin $n/2$ representation; odd $n$ gives the half-integer spins of electrons and other fermions.\nOn the Lie algebra, $V_n$ is the irreducible representation of $\\mathfrak{sl}_2(\\C)$ of highest weight $n$.\n\n## Explanations\n\n### L1 human\n\nThe group $\\mathrm{SU}(2)$ is the group of rotations of quantum spin.\nIts basic ways of acting on vector spaces are labelled by one number $n = 0, 1, 2, \\dots$, and the $n$-th one has dimension $n + 1$.\nThey are built from polynomials of degree $n$ in two variables.\nEvery other finite-dimensional representation is a sum of these basic ones.\n",
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