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  "body": "\n## Summary\n\nThe flag variety of a vector space $V$ parametrizes chains of subspaces $V_1 \\subset V_2 \\subset \\cdots$ of prescribed dimensions.\nIt is a smooth projective [[homogeneous space|85TWR6E75P]] for $\\mathrm{GL}(V)$; [[projective spaces|ZT9A6JK8QF]] and [[Grassmannians|CYPA882RB9]] are special cases.\nIn general, a flag variety is a quotient $G / P$ of a reductive group by a parabolic subgroup.\nFlag varieties are where representation theory, combinatorics and geometry meet, through [[Schubert varieties|Z014Q8KPDD]] and the Borel–Weil theorem.\n\n## Explanations\n\n### L1 human\n\nA flag in three-dimensional space is a line through the origin together with a plane through the origin that contains the line.\nThink of a flagpole and a flag attached to it.\nThe set of all such flags is a smooth, compact space of dimension $3$, the full flag variety of $\\C^3$.\nIn general a flag is a nested chain of subspaces, and the flag variety is the space of all of them.\n",
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