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  "body": "\n## Summary\n\nModular representation theory studies [[representations|WYGRXCG2RN]] of a finite group $G$ over a field whose characteristic $p$ divides the order of $G$.\nIn this case [[Maschke's theorem|4SBZ4EMPBZ]] fails, so representations need not split into irreducible pieces, and the [[group algebra|MNP2HBWB5Q]] is not semisimple.\nBrauer developed the theory, with tools such as Brauer characters, blocks, defect groups and the [[Cartan matrix|ZBQ4KRC98K]].\nIt links ordinary characters in characteristic $0$ with representations over finite fields through reduction modulo $p$.\n\n## Explanations\n\n### L1 human\n\nA representation lets a group act on a vector space by linear maps.\nOver the complex numbers, every representation of a finite group breaks into irreducible pieces, like a number into primes.\nOver a field such as $\\F_p$, where adding $p$ ones gives zero, this breaking can fail when $p$ divides the size of the group.\nFor the cyclic group of order $p$ acting on $\\F_p^2$ by $\\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix}$, the line spanned by the first basis vector is preserved, but no complementary line is.\nModular representation theory is the study of this richer and harder situation.\n",
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