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12
Appendix to “The Chow rings of generalized Grassmannians”
"... In this appendix we tabulate preliminary data required to establish Theorem 112 in the paper “The Chow rings of generalized Grassmannians”. This appendix is set to record the intermediate data required to establish Theorem 112 in [DZ3]. At the first sight, the data may be seen as many. However, th ..."
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In this appendix we tabulate preliminary data required to establish Theorem 112 in the paper “The Chow rings of generalized Grassmannians”. This appendix is set to record the intermediate data required to establish Theorem 112 in [DZ3]. At the first sight, the data may be seen as many. However, they are all produced (and are processed again) by computer. We begin with a brief guide for the content tabulated. Sections §1–§6 are devoted, respectively, to the six cases of (G, H) concerned by Theorem 16 (resp. Theorem 7–12) in [DZ3]. a) In 1.1–6.1, the sets W(H; G) (cf. [DZ3, 2.1]) of left cosets of the Weyl group of G by the Weyl group of H are presented both in terms of the minimized decompositions of its elements, and index system [DZ3,(2.3)] imposed by the decompositions. These are produced using the Decomposition in [DZ1]. b) In 1.2–6.2, the matrices Ak (cf. [DZ3, 4.3, Step 1]), k ≥ 1, required to determine the additive cohomology of G/Hs are listed. They are computed from the L–R coefficients in [DZ1]. c) In 1.3–6.3, the multiplicative rule for the basis elements of H even (G/Hs) are presented (cf. [DZ3, 4.3, Step 2]). They are also obtained by applying the L–R coefficients in [DZ1]. d) In 1.4–6.4, with respect to the ordered monomial basis B(m) (cf. [DZ3, 3.3]) for certain values of m, the structure matrices M(πm) are computed by using the L–R coefficients in [DZ1]. e) Applying the built–in function “Nullspace ” in Mathematica to the matrices M(πm) in 1.4–6.4, one obtains the Nullspace N(πm) listed in 1.5–6.5. These are required to specify the relations ri’s in the proofs of Theorem 1–6 in [DZ3,§6]. We conclude this introduction with the Cartan matrices for the Lie groups of types F4, E6 and E7 [Hu, p.59]. All data in this text are generated from them.
A description based on Schubert classes of cohomology of flag manifolds
, 2009
"... We describe the integral cohomology rings of the flag manifolds of types Bn, Dn, G2 and F4 in terms of their Schubert classes. The main tool is the divided difference operators of BernsteinGelfandGelfand and Demazure. As an application, we compute the Chow rings of the corresponding complex alg ..."
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We describe the integral cohomology rings of the flag manifolds of types Bn, Dn, G2 and F4 in terms of their Schubert classes. The main tool is the divided difference operators of BernsteinGelfandGelfand and Demazure. As an application, we compute the Chow rings of the corresponding complex algebraic groups, recovering thereby the results of R. Marlin.
THE CHOW RINGS OF THE ALGEBRAIC GROUPS E6 AND E7
"... Abstract. We determine the Chow rings of the complex algebraic groups E6 and E7, giving generators and relations in terms of Schubert classes of the corresponding flag varieties. This is a continuation of the work of R. Marlin on the computation of the Chow rings of SOn, Spin n, G2, and F4. ..."
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Abstract. We determine the Chow rings of the complex algebraic groups E6 and E7, giving generators and relations in terms of Schubert classes of the corresponding flag varieties. This is a continuation of the work of R. Marlin on the computation of the Chow rings of SOn, Spin n, G2, and F4.
The Chow rings of generalized Grassmanianns
, 2008
"... Based on the formula for multiplying Schubert classes obtained in [D2] and programed in [DZ1], we introduce a new method to compute the Chow ring of a flag variety G/H. As applications the Chow rings of some generalized Grassmannians G/H are presented as the quotients of polynomial rings in the spec ..."
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Based on the formula for multiplying Schubert classes obtained in [D2] and programed in [DZ1], we introduce a new method to compute the Chow ring of a flag variety G/H. As applications the Chow rings of some generalized Grassmannians G/H are presented as the quotients of polynomial rings in the special Schubert classes on G/H. 1
Samelson products in pregular exceptional Lie groups, available at arXiv:1403.4998
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A DESCRIPTION BASED ON SCHUBERT CLASSES OF COHOMOLOGY OF FLAG MANIFOLDS
, 709
"... Abstract. We describe the integral cohomology rings of the flag manifolds of types Bn, Dn, G2 and F4 in terms of their Schubert classes. The main tool is the divided difference operators of BernsteinGelfandGelfand and Demazure. As an application, we compute the Chow rings of the corresponding comp ..."
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Abstract. We describe the integral cohomology rings of the flag manifolds of types Bn, Dn, G2 and F4 in terms of their Schubert classes. The main tool is the divided difference operators of BernsteinGelfandGelfand and Demazure. As an application, we compute the Chow rings of the corresponding complex algebraic groups, recovering thereby the results of R. Marlin. 1.