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I was wondering if $G$ is a semisimple complex algebraic group, then is the homogeneous coordinate ring of a flag variety a UFD or not?

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    $\begingroup$ No, except for \emph{minimal} flag varieties (e.g. Grassmannians): as soon as $\operatorname{rk}\operatorname{Pic}(V)>1 $, the homogeneous coordinate ring of $V$ is not a UFD. See Hartshorne, exer. II.6.3.c). $\endgroup$
    – abx
    Commented Jun 23, 2022 at 16:28
  • $\begingroup$ For type A, see Fulton's Young Tableaux, Chapter 9. $\endgroup$
    – andrewBee
    Commented Jun 23, 2022 at 20:59

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