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  "body": "\n## Summary\n\nA Brauer character is the analogue of the [[character of a representation|RMWMYBY8TC]] for representations of a finite group over a field of prime characteristic $p$.\nIt is a complex-valued function on the elements of order prime to $p$.\nIts values come from lifting the eigenvalues of each matrix to complex roots of unity.\nBrauer characters determine the composition factors of a modular representation, which ordinary traces in characteristic $p$ cannot do.\nThey are the basic tool of modular representation theory.\n\n## Explanations\n\n### L1 human\n\nIn characteristic $p$ the trace of a matrix is only a number modulo $p$, and it loses information; for example the identity matrix of size $p$ has trace $0$.\nBrauer's idea is to replace each eigenvalue, a root of unity in the finite field world, by a matching complex root of unity, and add those up.\nThis gives a genuine complex number that behaves like an ordinary character.\nThe price is that it is only defined on elements whose order is not divisible by $p$.\n",
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