A. Borel, Sur la cohomologie des espaces fibr es principaux et des espaces homog nes de groupes de Lie compacts, Ann. of Math. 57 (1953), 115--207  Home/Search   Document Not in Database   Summary   Related Articles   Check

....DMS 9701755. 7. The Pieri type formula 38 7.1. Case ffi(i ) 1 40 7.2. Case ffi(i ) 0 41 References 45 Introduction The cohomology of a flag manifold G=B has an integral basis of Schubert classes Sw indexed by elements w of the Weyl group of G. The algebraic structure of these rings is known [9] with respect to a monomial basis, and there are methods (Schubert polynomials) for expressing the Sw in terms of this basis [6, 7, 12, 17, 19, 25, 29] Moreover, their multiplicative structure with respect to the Schubert basis is determined by Chevalley s formula [10] Despite this, it remains ....

....maximal isotropic subspaces of V , exhibiting it as the homogeneous space G=P 0 . Here P 0 is the stabilizer of a maximal isotropic subspace, a maximal parabolic subgroup associated to the simple root of exceptional length. Let : G=B i G=P 0 be the projection map. The rational cohomology rings [9] of G=B for both the symplectic and odd orthogonal flag manifolds are isomorphic to Q[x 1 ; x n ] he i (x n ) i = 1; ni; where e i (a 1 ; a n ) is the ith elementary symmetric polynomial in a 1 ; a n . However, their integral cohomology rings differ [14] ....

A. Borel, Sur la cohomologie des espaces fibr'es principaux et des espaces homog`enes des groupes de Lie compacts, Ann. Math., 57 (1953), pp. 115--207.

.... i n and E j = spanff 1 ; f j g; for 1 j ng; which has codimension (w) Its closure is the Schubert variety XwF : The cohomology classes [Xw F : of the Schubert varieties XwF : give an integral basis for the cohomology ring of the flag variety [8] Using Chern classes, Borel [5] gave Strictly speaking, we mean the classes Poincar e dual to the fundamental cycles in homology. an alternate description of this ring as H n = Z[x 1 ; x n ] S, where S is the ideal generated by the symmetric polynomials. These two descriptions were reconciled by Demazure [7] and ....

A. Borel, Sur la cohomologie des espaces fibr'es principaux et des espaces homog`enes des groupes de Lie compacts, Ann. Math., 57 (1953), pp. 115--207.

....: s n (that is, the symmetric group Sn 1 ) Let x 1 , x 2 , be formal variables. Then W naturally acts on the polynomial ring C [x 1 ; xn 1 ] by permuting the variables. Let I W denote the ideal generated by homogeneous non constant W symmetric polynomials. By a classical result [Bo], the cohomology ring H(F ) of the flag variety of type A can be canonically identified with the quotient C [x 1 ; xn 1 ] I W . This ring is graded by the degree and has a distinguished linear basis of homogeneous cosets Xw modulo I W , labelled by the elements w of the group. Let us state ....

A. Borel, Sur la cohomologie des espaces fibr'es principaux et des espaces homog'enes des groupes de Lie compacts, Ann. Math. 57 (1953), 115-207.

....j = f0g for all j. The set of all flags is an dimensional complex manifold, F V (or F n ) called the flag manifold. There is a tautological flag mathcalF q of bundles over F V whose fibre at F q is F q . Let Gammax i be the first Chern class of the line bundle F i =F i Gamma1 . Borel [10] showed the cohomology ring of F V to be Z[x 1 ; x n ] he i (x 1 ; x n ) j i = 1; ni; where e i (x 1 ; x n ) is the ith elementary symmetric polynomial in x 1 ; x n . Given a subset S ae V , let hSi be its linear span. For subspaces W ae U , let U Gamma ....

A. Borel, Sur la cohomologie des espaces fibr'es principaux et des espaces homog`enes des groupes de Lie compacts, Ann. Math., 57 (1953), pp. 115--207.

....is the flag F q . Let x i be the Chern class of the line bundle F i =F i Gamma1 . Then the integral cohomology ring of ) is H n = 1 ; x n ] S, where S is the ideal generated by those non constant polynomials which are symmetric in x 1 ; x n . This description is due to Borel [5]. Given a subset S ae V , let hSi be its linear span and for linear subspaces W ae U let U Gamma W be their set theoretic difference. An ordered basis f 1 ; f 2 ; f n for V determines a flag E q ; set E i = hf 1 ; f i i for 1 i n. In this case, write E = hf 1 ; f n i ....

A. Borel, Sur la cohomologie des espaces fibr'es principaux et des espaces homog`enes des groupes de Lie compacts, Ann. Math., 57 (1953), pp. 115--207.

.... F j = f0g for all j. The set of all flags is an dimensional complex manifold, F V (or F n ) called the flag manifold. There is a tautological flag F q of bundles over F V whose fibre at F q is F q . Let Gammax i be the first Chern class of the line bundle F i =F i Gamma1 . Borel [11] showed the cohomology ring of F V to be Z[x 1 ; x n ] he i (x 1 ; x n ) j i = 1; ni; where e i (x 1 ; x n ) is the ith elementary symmetric polynomial in x 1 ; x n . Given a subset S ae V , let hSi be its linear span. For subspaces W ae U , let U Gamma ....

A. Borel, Sur la cohomologie des espaces fibr'es principaux et des espaces homog`enes des groupes de Lie compacts, Ann. Math., 57 (1953), pp. 115--207.

....theory of Schubert polynomials. We also derive the quantum Monk s formula. 1. Introduction Let F l n be the manifold of complete flags in the n dimensional linear space C n . The cohomology ring H (F l n ; Z) can be described in two different ways. An algebraic description due to A. Borel [5] represents it canonically as a quotient of a polynomial ring: H (F l n ; Z) Z[x 1 ; xn ] I n ; 1.1) where I n is the ideal generated by symmetric polynomials in x 1 ; xn without constant term. Another, geometric, description of the cohomology ring of the flag manifold is ....

....the line bundle E i =E i Gamma1 . Let I n ae Z[x 1 ; xn ] be the ideal generated by all symmetric polynomials without constant term or, equivalently, by the elementary symmetric polynomials e i (x 1 ; xn ) for i = 1; n. The following classical result is due to A. Borel [5]. 1 Note added in proof. The latest developments in quantum Schubert calculus are reviewed and unified in W. Fulton s recent note Universal Schubert polynomials . QUANTUM SCHUBERT POLYNOMIALS 569 Theorem 2.1. The kernel of the homomorphism is I n . The induced map Z[x 1 ; xn ] I ....

A. Borel, Sur la cohomologie des espaces fibr'es principaux et des espaces homog`enes des groupes de Lie compacts, Ann. of Math. (2) 57 (1953), 115--207. MR 14:490e

....it is not easy to see how to use the Serre spectral sequence associated to the fibration 8.4 to compute the two key groups H 2n Gamma3 (S(p) Z) and H 2n Gamma2 (S(p) Z) Thus, to prove theorem 8. 1 we use a spectral sequence argument of Eschenburg [Esch] which exploits ideas of Borel [Bor1]. To begin let M be a compact manifold and U a compact Lie Group that acts freely on M . Further assume that the cohomology rings of both M and U are known and one wants to compute the cohomology of M=U: Rather than using the principal U bundle 8:5 U Gamma Gamma M Gamma Gamma M=U analogous ....

....G = U(n) has E ; 2 ( Delta) H (B U(n) 2 ; Z) Omega H (U(n) Z) as the cohomology of the base and fibre are torsion free. Let k j : H (B U(n) 2 ; Z) Gamma Gamma E ;0 j ( Delta) denote the natural projection of the E ;0 2 ( Delta) term along the base. Proposition 8. 10: [Bor1] Delta = k1 : H (B U(n) 2 ; Z) Gamma Gamma E ;0 1 ( Delta) ae H (U(n) Z) Using this proposition of Borel and the fact that Delta induces the cup product in H (BG ) so Delta (u Omega 1) Delta (1 Omega u) u; Eschenburg computed the differentials in E ; j ....

A. Borel, Sur la cohomologie des espaces fibr'es principaux et des espaces homog'enes de groupes de Lie compacts, Ann. of Math., 57 (1953), 115-207.

....of 8.2 has the form E ; 2 ( Delta) H (B U(n) 2 ; Z) Omega H (U(n) Z) as the cohomology of the base and fibre are torsion free. Let k j : H (B U(n) 2 ; Z) Gamma Gamma E ;0 j ( Delta) denote the natural projection of the E ;0 2 ( Delta) term along the base. Proposition 8. 4: [Bor1] Delta = k1 : H (B U(n) 2 ; Z) Gamma E ;0 1 ( Delta) ae H (U(n) Z) Using this proposition of Borel and the fact that Delta induces the cup product in H (BG ) so Delta (u Omega 1) Delta (1 Omega u) u; Eschenburg computed the differentials in E ; j ( Delta) Lemma 8.5: ....

A. Borel, Sur la cohomologie des espaces fibr'es principaux et des espaces homog'enes de groupes de Lie compacts, Ann. of Math., 57 (1953), 115-207.

....colimit. However, one must be quite careful about basepoints, and about what happens when ff 1 (c) is not abelian. These points are handled in detail by Wojtkowiak in [Wo] Theorem 1.2 is a special case of the following spectral sequence, constructed by Bousfield Kan ( BK, XII.4.1 XI.7. 1] and [Bo]) As we will show later, this can often be used for explicit calculations involving R Gamma1 ( f) even when the obstructions of Theorem 1.2 do not all vanish. Theorem 1.3. Let F , R, and f be as in Theorem 1.2. Let map(hocolim Gamma (F ) X) f be the union of the components of the ....

A. Borel, Sur la cohomologie des espaces fibr'es principaux et des espaces homog `enes de groupes de Lie compacts, Ann. Math. 57 (1953), 115--207

....SOTTILE 7. The Pieri type formula 37 7.1. Case ffi(i ) 1 40 7.2. Case ffi(i ) 0 40 References 45 Introduction The cohomology of a flag manifold G=B has an integral basis of Schubert classes Sw indexed by elements w of the Weyl group of G. The algebraic structure of these rings is known [9] with respect to a monomial basis, and there are methods (Schubert polynomials) for expressing the Sw in terms of this basis [6, 7, 12, 17, 19, 25, 29] Moreover, their multiplicative structure with respect to the Schubert basis is determined by Chevalley s formula [10] Despite this, it remains ....

....maximal isotropic subspaces of V , exhibiting it as the homogeneous space G=P 0 . Here P 0 is the stabilizer of a maximal isotropic subspace, a maximal parabolic subgroup associated to the simple root of exceptional length. Let : G=B i G=P 0 be the projection map. The rational cohomology rings [9] of G=B for both the symplectic and odd orthogonal flag manifolds are isomorphic to Q [x 1 ; x n ] he i (x 2 1 ; x 2 n ) i = 1; ni; where e i (a 1 ; a n ) is the ith elementary symmetric polynomial in a 1 ; a n . However, their integral cohomology ....

A. Borel, Sur la cohomologie des espaces fibr'es principaux et des espaces homog`enes des groupes de Lie compacts, Ann. Math., 57 (1953), pp. 115--207.

....55, Odense M, DK 5230, Denmark. E mail: henrik imada.ou.dk. z Address: Department of Mathematics, University of California at Riverside, Riverside, CA 92521, U.S.A. E mail: ypoon math.ucr.edu. Joyce s construction of homogeneous hypercomplex structures has its roots in the work of Borel [1], Samelson [17] and Wang [19] about homogeneous complex manifolds. It is also tied to Wolf s construction of symmetric quaternionic Kahler manifolds [20] Suppose that G is a compact semi simple Lie group of rank r. Joyce and Spindel et al. found that the group T 2n Gammar Theta G has a ....

A. Borel. Sur la cohomologie des espaces fibr'es principaux et des espaces homog'enes de groupes de Lie compacts, Ann. Math. 57 (1953) 115--207.

....Schubert basis. The c w u v are non negative: They enumerate flags in a suitable triple intersection of Schubert varieties. Evaluating a Schubert polynomial at certain Chern classes gives a Schubert class in the cohomology of the flag manifold. This exhibits the cohomology of the flag manifold [10] as: Z[x 1 ; x 2 ; hS w j w 62 S n i: It remains an open problem to give a bijective formula for these constants. We expect such a formula will have the form c w u v = # ae (saturated) chains in the Bruhat order on S1 from u to w satisfying some condition imposed by v oe : 1) ....

....for all j. The set of all flags is an Gamma n 2 Delta dimensional complex manifold, F V (or F n ) called the flag manifold. There is a tautological flag F q of bundles over F V whose fibre at F q is F q . Let Gammax i be the first Chern class of the line bundle F i =F i Gamma1 . Borel [10] showed the cohomology ring of F V is Z[x 1 ; x n ] he i (x 1 ; x n ) j i = 1; ni; where e i (x 1 ; x n ) is the ith elementary symmetric polynomial in x 1 ; x n . Let hSi be the linear span of S ae V and U Gamma W be the set theoretic difference of ....

A. Borel, Sur la cohomologie des espaces fibr'es principaux et des espaces homog`enes des groupes de Lie compacts, Ann. Math., 57 (1953), pp. 115--207.

....F q . Let Gammax i be the Chern class of the line bundle F i =F i Gamma1 . Then the integral cohomology ring of F(V ) is H n = Z[x 1 ; x n ] S, where S is the ideal generated by those non constant polynomials which are symmetric in x 1 ; x n . This description is due to Borel [5]. Given a subset S ae V , let hSi be its linear span and for linear subspaces W ae U let U Gamma W be their set theoretic difference. An ordered basis f 1 ; f 2 ; f n for V determines a flag E q ; set E i = hf 1 ; f i i for 1 i n. In this case, write E q = hf 1 ; f n ....

A. Borel, Sur la cohomologie des espaces fibr'es principaux et des espaces homog`enes des groupes de Lie compacts, Ann. Math., 57 (1953), pp. 115--207.

....by the identity S u Delta S v = X w c w u v Sw : These c w u v are non negative as they are the intersection multiplicity of a suitable triple intersection of Schubert varieties. There exist algorithms for computing these numbers c w u v : The algebraic structure of these rings is known [9] with respect to a monomial basis, and there are methods (Schubert polynomials) for expressing the Sw in terms of this basis [6, 7, 11, 14, 16, 22, 29] However, these algorithms do not show the non negativity of the c w u v . When S v is a hypersurface Schubert class, the c w u v are either ....

....maximal isotropic subspaces of V , exhibiting it as the homogeneous space G=P 0 . Here P 0 is the stabilizer of a maximal isotropic subspace, a maximal parabolic subgroup associated to the simple root of exceptional length. Let : G=B i G=P 0 be the projection map. The rational cohomology rings [9] of G=B for both the symplectic and oddorthogonal flag manifolds are isomorphic to Q [x 1 ; x n ] he i (x 2 1 ; x 2 n ) i = 1; ni ; where e i (a 1 ; a n ) is the ith elementary symmetric polynomial in a 1 ; a n . However, their integral cohomology ....

A. Borel, Sur la cohomologie des espaces fibr'es principaux et des espaces homog`enes des groupes de Lie compacts, Ann. Math., 57 (1953), pp. 115--207.

....r eseau 32 7. The Pieri type formula 34 7.1. Case ffi(i ) 1 36 7.2. Case ffi(i ) 0 37 References 41 Introduction The cohomology of a flag manifold G=B has an integral basis of Schubert classes Sw indexed by elements w of the Weyl group of G. The algebraic structure of these rings is known [9] with respect to a monomial basis, and there are methods (Schubert polynomials) for expressing the Sw in terms of this basis [6, 7, 12, 17, 19, 25, 29] Moreover, their multiplicative structure with respect to the Schubert basis is determined by Chevalley s formula [10] Despite this, it remains ....

....maximal isotropic subspaces of V , exhibiting it as the homogeneous space G=P 0 . Here P 0 is the stabilizer of a maximal isotropic subspace, a maximal parabolic subgroup associated to the simple root of exceptional length. Let : G=B i G=P 0 be the projection map. The rational cohomology rings [9] of G=B for both the symplectic and odd orthogonal flag manifolds are isomorphic to Q [x 1 ; x n ] he i (x 2 1 ; x 2 n ) i = 1; ni; PIERI TYPE FORMULA 19 where e i (a 1 ; a n ) is the ith elementary symmetric polynomial in a 1 ; a n . However, their ....

A. Borel, Sur la cohomologie des espaces fibr'es principaux et des espaces homog`enes des groupes de Lie compacts, Ann. Math., 57 (1953), pp. 115--207.

....: s n (that is, the symmetric group Sn 1 ) Let x 1 , x 2 , be formal variables. Then W naturally acts on the polynomial ring C [x 1 ; xn 1 ] by permuting the variables. Let I W denote the ideal generated by homogeneous non constant W symmetric polynomials. By a classical result [Bo], the cohomology ring H(F ) of the flag variety of type A can be canonically identified with the quotient C [x 1 ; xn 1 ] I W . This ring is graded by the degree and has a distinguished linear basis of homogeneous cosets Xw modulo I W , labelled by the elements w of the group. Let us ....

A. Borel, Sur la cohomologie des espaces fibr'es principaux et des espaces homog'enes des groupes de Lie compacts, Ann. Math. 57 (1953), 115-207.

.... only the following groups H i (N 2 ) are nontrivial: H 0 H 5 Z; H 3 Z 2 : This statement is not new: indeed, it is easy to see that the space Pi 2 n Sigma 2 is homotopy equivalent to the Lagrangian Grassmannian manifold U(3) O(3) whose cohomology groups were studied in [4] and [6] see e.g. 10] We give here another calculation, demonstrating our general method. This calculation is based on the classification of subsets in CP 2 , which can be the singular sets of homogeneous polynomials of degree 2 in C 3 : There are exactly the following such sets: A) any ....

A. Borel. Sur la cohomologie des espaces fibr'es principaux et des espaces homog `enes de groupes de Lie compacts. Ann. of Math. 1953 57 (2), 115--207.

.... i n and E j = spanff 1 ; f j g; for 1 j ng; which has codimension (w) Its closure is the Schubert variety XwF : The cohomology classes 1 [Xw F : of the Schubert varieties XwF : give an integral basis for the cohomology ring of the flag variety [8] Using Chern classes, Borel [5] gave 1 Strictly speaking, we mean the classes Poincar e dual to the fundamental cycles in homology. 4 FRANK SOTTILE an alternate description of this ring as H n = Z[x 1 ; x n ] S, where S is the ideal generated by the symmetric polynomials. These two descriptions were reconciled by ....

A. Borel, Sur la cohomologie des espaces fibr'es principaux et des espaces homog`enes des groupes de Lie compacts, Ann. Math., 57 (1953), pp. 115--207.

....be the derivation of P Omega s Phi determined by choosing a splitting Phi I and factoring it as ffis : Phi s Phi I: In terms of representatives ae 2 Phi; ffiae is s Gamma1 ae. In other words, P Omega s Phi is the Koszul complex for the ideal I in the commutative algebra P [Ko] [Bo]. If I is what is now known as a regular (at one time: Borel) ideal (an algebraic condition, but implied by I being the defining ideal in C 1 (W ) for V = J Gamma1 (0) when 0 is a regular value of J : W R N ) the Koszul complex (P Omega s Phi; ffi) is a model for P=I. For more general ....

A. Borel, Sur la cohomologie des espaces fibr'es principaux et des espaces homogenes de groupes de Lie compacts, Annals of Math. 57 (1953) 115-207.

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A. Borel, Sur la cohomologie des espaces fibr es principaux et des espaces homog nes de groupes de Lie compacts, Ann. of Math. 57 (1953), 115--207

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A. Borel, Sur la cohomologie des espaces fibr'es principaux et des espaces homog`enes des groupes de Lie compacts, Ann. of Math. 57 (1953), 322--391.

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A. Borel, Sur la cohomologie des espaces fibr'es principaux et des espaces homog`enes de groupes de Lie compacts, Annals of Math., 57(1953), 115--207.

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A. Borel. "Sur la cohomologie des espaces fibr'es principaux et des espaces homog`enes de groupes de Lie compacts." Ann. Math. 57 (1953), 115--207.

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Borel, A.. Sur la cohomologie des espaces fibr'es principaux et des espaces homog`enes des groupes de Lie compacts. Annals of Math., 57 (1953), 115-207.

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