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I've recently read the following line in an interesting paper:

It is well-known that the cohomology ring of a flag variety $G/B$ is isomorphic to the quotient ring of the ring of polynomial functions on the Cartan sub algebra $\frak{h}$ by the ideal generated by the fundamental invariants $f_1 , . . . , f_r, r = $rank$({\frak h})$, of the Weyl group W, i.e. $$ H^∗(G/B,{Q}) \simeq {\text Sym}_Q{\frak h}^∗/(f_1, . . . , f_r). $$

I would like to ask:

(1) Does this result extend to other fields, i.e. the complex and real case?

(2) What is a good understandable reference for learning about this result?

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    $\begingroup$ There is a lot of related discussion along with references in a previous question 21651. In your question (2), the word "understandable" of course means different things to different people, so it's good to look at various treatments. $\endgroup$
    – Jim Humphreys
    Commented Nov 7, 2012 at 14:46
  • $\begingroup$ P.S. Keep in mind that the concrete treatments by Fulton and Manivel focus on the most classical framework of Lie type $A$, where the Weyl group is a symmetric group, while Borel (and Hiller) give a more comprehensive treatment in the language of Lie groups or algebraic groups and Weyl groups. $\endgroup$
    – Jim Humphreys
    Commented Nov 7, 2012 at 18:11
  • $\begingroup$ Jim's link: mathoverflow.net/questions/21651 $\endgroup$
    – Allen Knutson
    Commented Nov 8, 2012 at 3:09

1 Answer 1

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(1) Assuming you are referring to the coefficient field for your cohomology theory, then yes, the result immediately extends to $\mathbb{R}$ and $\mathbb{C}$ coefficients.

(2) Two sources are Fulton's Young Tableaux (Chapters 9 and 10) and Manivel's Symmetric Functions, Schubert Polynomials, and Degeneracy Loci (Chapter 3). Both of these sources assume a basic knowledge of and familiarity with algebraic geometry.

Edit: The original source for the result is a paper of Armand Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math. (2) 57, (1953), 115–207.

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