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  "body": "\n## Statement\n\nLet $G$ be a [[compact group|EVHCE5VR6Y]] with normalised [[Haar measure|VY0HXD3GC4]] $dg$, and for $\\alpha, \\beta \\in L^2(G)$ write $\\ip{\\alpha}{\\beta} = \\int_G \\alpha(g)\\overline{\\beta(g)} \\, dg$.\nLet $\\pi$ and $\\sigma$ be finite-dimensional continuous [[irreducible|FX3NEFYFFN]] [[unitary representations|74BPYJNZTM]] of $G$, of dimension $d_\\pi$ and $d_\\sigma$.\nLet $\\pi_{ij}$ and $\\sigma_{kl}$ be their matrix coefficients in orthonormal bases.\n\n1. If $\\pi$ and $\\sigma$ are not isomorphic, then $\\ip{\\pi_{ij}}{\\sigma_{kl}} = 0$ for all indices.\n2. For all indices $i, j, k, l$, one has $\\ip{\\pi_{ij}}{\\pi_{kl}} = \\delta_{ik}\\delta_{jl}/d_\\pi$.\n3. The [[characters|RMWMYBY8TC]] $\\chi_\\pi = \\tr \\pi$ and $\\chi_\\sigma = \\tr \\sigma$ satisfy $\\ip{\\chi_\\pi}{\\chi_\\sigma} = 1$ if $\\pi \\cong \\sigma$ and $\\ip{\\chi_\\pi}{\\chi_\\sigma} = 0$ otherwise.\n\n## Remarks\n\nFor a linear map $A$ between the two spaces, the average $\\int_G \\sigma(g) A \\pi(g)^{-1} \\, dg$ intertwines $\\pi$ and $\\sigma$.\nBy Schur's lemma it is $0$ when they are not isomorphic and a scalar when $\\pi = \\sigma$.\nTaking $A$ to be a matrix unit gives parts 1 and 2, the scalar being found by taking traces; summing over $i = j$ and $k = l$ gives part 3.\nExample: for $G = \\mathbb{T}$ with the characters $\\chi_n(z) = z^n$, part 3 reads $\\frac{1}{2\\pi}\\int_0^{2\\pi} e^{in\\theta} e^{-im\\theta} \\, d\\theta = \\delta_{nm}$.\nFor a finite group with Haar measure the counting measure divided by $\\abs{G}$, the statement is the [[Schur orthogonality relations|AY0B20QC04]] of finite groups.\nA consequence: a finite-dimensional continuous representation with character $\\chi$ is irreducible if and only if $\\ip{\\chi}{\\chi} = 1$.\n\n## Explanations\n\n### L1 human\n\nThe matrix entries of different irreducible representations of a compact group are perpendicular to each other as functions on the group.\nThe characters of irreducible representations are perpendicular unit vectors.\nSo one can find how often an irreducible piece occurs in a representation by an integral, just like taking a coordinate.\n\n### L2 human\n\nThese relations give the orthonormality half of the [[Peter–Weyl theorem|6VC9QE39XH]]; the completeness half is the harder part.\nFor compact connected Lie groups, the [[Weyl integration formula|QMH530N27M]] turns the character relation into an integral over a maximal torus, which is the route to the [[Weyl character formula|230F5FXPMQ]].\nThe relations apply to every finite-dimensional continuous representation after a change of basis, since each one is unitary for a suitable inner product by [[complete reducibility|N6JC6KNXJJ]].\n",
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