This is necessary to obtain related cohomology valued series for given arbitrary vector bundle and multiplicative characteristic class. Moreover, this can be used as a valuable testing ground for the theories which associates quantum cohomologies and J functions of non-abelian quotient to abelian quotients via quantization. | Turk J Math 32 (2008) , 61 – 82. ¨ ITAK ˙ c TUB Mirror Principle for Flag Manifolds Vehbi Emrah Paksoy Abstract In this paper, using mirror principle developped by Lian, Liu and Yau [8, 9, 10, 11, 12, 13] we obtained the A and B series for the equivariant tangent bundles over homogenous spaces using Chern polynomial. This is necessary to obtain related cohomology valued series for given arbitrary vector bundle and multiplicative characteristic class. Moreover, this can be used as a valuable testing ground for the theories which associates quantum cohomologies and J functions of non-abelian quotient to abelian quotients via quantization.∗ 1. Introduction It is an interesting question to obtain A series for equivarant tangent bundles and Chern Polynomials since this will be necessary to obtain A series for a general vector bundle and multiplicative characteristic class. Now assume T is an algebraic torus and X be a T-manifold with a T equivariant embedding in Y := Pm1 × · · · × Pml such that pull backs of hyperplane classes H = (H1 , . . . , Hl ) generate H 2 (X, Q). We will use the same ˇ ⊂ H2 (X) be the set notations for equivariant classes and their restriction to X. Let K ˇ is a semiahler cone of X. K of points in H2 (X, Z)free in the dual of the closure of the K¨ ˇ group and defines a partial ordering on H2 (X, Q)free . Explicitly r d iff d − r ∈ K. ˇ ˇ ˇ If {Hj } is the dual basis for {Hi } in H2 (X), r d ⇔ d − r = d1 H1 + · · · + dl Hl where di , i = 1, . . . , l are nonnegative integers. Let X = F l(n) be the complete flag variety. The first Chow ring A1 (X) ∼ = H 2 (X, Z) is generated by Si = c1 (Lλi ), i = 1, . . . , n − 1 and λi is the dominant weight of torus action with λi = (1, . . . , 1, 0, . . . , 0) first i terms are ∗I would like to thank Bong H. Lian for his precious helps and guideance. 61 PAKSOY 1’s. Here, Lλi is the line bundle over X, associated the 1 dimensional representation with respect to weight λi . For more on homogenous .
