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  "body": "\n## Statement\n\nLet $G$ be a finite [[group|G5S1M9]].\nFor functions $\\alpha, \\beta \\colon G \\to \\C$ write\n$$\\ip{\\alpha}{\\beta} = \\frac{1}{\\abs{G}} \\sum_{g \\in G} \\alpha(g) \\overline{\\beta(g)},$$\nwhere the bar is the [[complex conjugate|0FTH2E9TAS]].\n\n1. Let $\\rho$ and $\\sigma$ be [[irreducible|FX3NEFYFFN]] complex representations of $G$ by [[unitary matrices|0AR68JTMDT]], of degrees $d_\\rho$ and $d_\\sigma$. If $\\rho$ and $\\sigma$ are not isomorphic, then $\\ip{\\rho_{ij}}{\\sigma_{kl}} = 0$ for all indices. For all indices $i, j, k, l$ one has $\\ip{\\rho_{ij}}{\\rho_{kl}} = \\delta_{ik}\\delta_{jl} / d_\\rho$.\n2. Let $\\chi$ and $\\psi$ be irreducible complex [[characters|RMWMYBY8TC]] of $G$. Then $\\ip{\\chi}{\\psi} = 1$ if $\\chi = \\psi$ and $\\ip{\\chi}{\\psi} = 0$ otherwise.\n3. Let $g, h \\in G$, and let $C_G(g)$ be the [[centraliser|02YYV36BFM]] of $g$. Then $\\sum_{\\chi} \\chi(g) \\overline{\\chi(h)}$, summed over all irreducible complex characters $\\chi$ of $G$, equals $\\abs{C_G(g)}$ if $g$ and $h$ are [[conjugate|0DEBXAJ5SA]], and $0$ otherwise.\n\n## Remarks\n\nThe proof of the first relation applies Schur's lemma to the averaged map $\\sum_{g} \\rho(g) A \\sigma(g)^{-1}$, which commutes with the action for every matrix $A$.\nTaking traces in the first relation gives the second.\nThe second relation shows that the irreducible characters are orthonormal in the space of class functions; they span this space, and the third relation follows from that.\nFor example, for $G = \\Z/2\\Z$ the two irreducible characters are $(1, 1)$ and $(1, -1)$, and $\\ip{\\chi}{\\psi} = \\tfrac{1}{2}(1 - 1) = 0$.\nA consequence: a complex representation with character $\\chi$ is irreducible if and only if $\\ip{\\chi}{\\chi} = 1$.\nEvery complex representation of a finite group is isomorphic to a unitary one, so the first relation applies to all irreducible representations after a change of basis.\n\n## Explanations\n\n### L1 human\n\nThe irreducible characters of a finite group behave like perpendicular unit vectors.\nIf one averages the product of two different irreducible characters over the group, the result is zero.\nThis makes it easy to decompose any representation: the number of times an irreducible piece occurs is an inner product, like a coordinate along an axis.\n\n### L2 human\n\nThe relations are the main computational tool of [[character tables|P16AFZT3K0]].\nRow orthogonality (relation 2) gives the multiplicity $\\ip{\\chi_V}{\\chi}$ of an irreducible $\\chi$ in a representation $V$.\nColumn orthogonality (relation 3) with $g = h = 1$ gives $\\sum_\\chi \\chi(1)^2 = \\abs{G}$.\nTogether they show that the number of irreducible characters equals the number of conjugacy classes, and they let one fill in missing entries of a character table.\nThey are the finite group case of the Peter–Weyl theory for compact groups, where the average becomes an integral against Haar measure.\n",
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