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  "body": "\n## Summary\n\nThe character of a finite-dimensional [[group representation|WYGRXCG2RN]] $\\rho$ is the function $g \\mapsto \\tr \\rho(g)$ on the group.\nIt is constant on [[conjugacy classes|0DEBXAJ5SA]], because the trace does not change under conjugation.\nFor a finite group over the complex numbers, a representation is determined up to isomorphism by its character.\nSo characters turn the study of representations into the study of a finite list of numbers.\n\n## Explanations\n\n### L1 human\n\nReplace each matrix of a representation by one number, its trace, the sum of its diagonal entries.\nSurprisingly little is lost: for finite groups over $\\C$ these numbers already tell which representation one has.\nFor example, let the symmetric group $S_3$ permute the coordinates of $\\C^3$.\nA permutation matrix has trace equal to the number of points it fixes.\nSo the character takes the value $3$ at the identity, $1$ at each transposition and $0$ at each $3$-cycle.\n",
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