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  "body": "\n## Statement\n\nLet $G$ be a connected [[compact Lie group|9TFC6ADC1C]], and let $T$ be a [[maximal torus|3JVGZVVTB0]] with Lie algebra $\\mathfrak{t}$.\nLet $W$ be the [[Weyl group|EZGTEK3HQN]], acting on the dual space $\\mathfrak{t}^*$.\nIdentify each continuous character of $T$ with the linear form $\\mu \\in \\mathfrak{t}^*$ such that the character sends $\\exp H$ to $e^{i\\mu(H)}$ for $H \\in \\mathfrak{t}$.\nLet $\\Phi \\subset \\mathfrak{t}^*$ be the roots of $G$, choose a set $\\Phi^+$ of positive roots, and let $\\rho = \\frac{1}{2}\\sum_{\\alpha \\in \\Phi^+} \\alpha$.\nFor $w \\in W$ let $\\varepsilon(w) = \\det(w) \\in \\set{1, -1}$.\nLet $V$ be a finite-dimensional [[irreducible|FX3NEFYFFN]] continuous representation of $G$ with highest weight $\\lambda$, and let $\\chi_V$ be its [[character|RMWMYBY8TC]].\nThen for every $H \\in \\mathfrak{t}$ with $\\prod_{\\alpha \\in \\Phi^+} \\sin(\\alpha(H)/2) \\ne 0$,\n$$\\chi_V(\\exp H) = \\frac{\\sum_{w \\in W} \\varepsilon(w)\\, e^{i (w(\\lambda + \\rho))(H)}}{\\sum_{w \\in W} \\varepsilon(w)\\, e^{i (w\\rho)(H)}}.$$\n\n## Remarks\n\nThe denominator equals $\\prod_{\\alpha \\in \\Phi^+} \\left(e^{i\\alpha(H)/2} - e^{-i\\alpha(H)/2}\\right)$, by the Weyl denominator formula.\nOne proof multiplies $\\chi_V$ by the denominator and notes that the product is $W$-alternating with integer coefficients.\nThe [[Weyl integration formula|QMH530N27M]] and $\\int_G \\abs{\\chi_V}^2 \\, dg = 1$ then show that only the term $\\lambda + \\rho$ and its $W$-images survive, each with coefficient $\\pm 1$.\nExample: for $G = SU(2)$ and $H = \\operatorname{diag}(i\\theta, -i\\theta)$, there is one positive root with $\\alpha(H) = 2\\theta$, $\\rho(H) = \\theta$, and the representation of dimension $n + 1$ has $\\lambda(H) = n\\theta$.\nThe formula gives $\\chi(\\exp H) = \\frac{e^{i(n+1)\\theta} - e^{-i(n+1)\\theta}}{e^{i\\theta} - e^{-i\\theta}} = \\frac{\\sin((n+1)\\theta)}{\\sin\\theta}$, which is $e^{in\\theta} + e^{i(n-2)\\theta} + \\cdots + e^{-in\\theta}$.\nLetting $H \\to 0$ gives the Weyl dimension formula $\\dim V = \\prod_{\\alpha \\in \\Phi^+} \\frac{\\ip{\\lambda + \\rho}{\\alpha}}{\\ip{\\rho}{\\alpha}}$ for a $W$-invariant inner product on $\\mathfrak{t}^*$.\n\n## Explanations\n\n### L1 human\n\nThe character of a representation records the trace of each group element, and it determines the representation.\nThe Weyl formula gives this character in closed form, as a ratio of two alternating sums of exponentials.\nOnly the highest weight $\\lambda$ of the representation enters, together with data of the group: its roots and its Weyl group.\n\n### L2 human\n\nBy [[conjugacy of maximal tori|BE3XK5ERC1]], a character is determined by its values on $T$, so the formula gives it everywhere.\nTogether with the [[Peter–Weyl theorem|6VC9QE39XH]] it gives the theorem of the highest weight: irreducible representations of $G$ correspond to dominant integral weights.\nThe dimension formula and Kostant's multiplicity formula follow from it, and the [[Borel–Weil–Bott theorem|K10YADF5XV]] gives a geometric realisation of the same representations.\nFor $G = U(n)$ the formula expresses irreducible characters as Schur polynomials in the eigenvalues.\n",
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