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  "body": "\n## Definition\n\nLet $k$ be a field, let $V$ be a vector space of dimension $n$ over $k$, and let $0 \u003c d_1 \u003c \\cdots \u003c d_m \u003c n$ be integers.\nA *flag* of type $(d_1, \\ldots, d_m)$ in $V$ is a chain of subspaces $V_1 \\subset \\cdots \\subset V_m$ with $\\dim V_i = d_i$.\nThe *flag variety* $\\mathrm{Fl}(d_1, \\ldots, d_m; V)$ is the set of these flags.\nIt is a closed subvariety of the product of [[Grassmannians|CYPA882RB9]] $\\prod_i \\mathrm{Gr}(d_i, V)$, so it is a [[projective variety|A4H4CZ2V35]].\nThe *full flag variety* is the case $(d_1, \\ldots, d_m) = (1, 2, \\ldots, n - 1)$.\n\n## Remarks\n\nThe group $\\mathrm{GL}(V)$ acts transitively on the flags of a given type, so the flag variety is a [[homogeneous space|85TWR6E75P]].\nThe stabilizer of a full flag is a [[Borel subgroup|5JSYA1JNCC]], conjugate to the upper triangular matrices, so the full flag variety is $\\mathrm{GL}_n / B$.\n\nGeneral form: let $G$ be a connected [[reductive group|2XS84PWE3S]] over an algebraically closed field.\nA closed subgroup $P$ of $G$ is *parabolic* if it contains a Borel subgroup.\nThe *generalized flag variety* is $G / P$, a smooth projective homogeneous variety.\n\nExample: for $m = 1$, the flag variety $\\mathrm{Fl}(d_1; V)$ is the Grassmannian $\\mathrm{Gr}(d_1, V)$, and for $d_1 = 1$ it is the projective space $\\mathbb{P}(V)$.\n\nNon-example: let $T$ be the diagonal torus of $\\mathrm{GL}_2$, the stabilizer of the pair of coordinate lines.\nThe quotient $\\mathrm{GL}_2 / T$ is the space of ordered pairs of distinct lines in $k^2$, which is $\\mathbb{P}^1 \\times \\mathbb{P}^1$ minus its diagonal.\nIt is not projective, since $T$ contains no Borel subgroup, and it is not a flag variety.\n",
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