Equivariant Cohomology in Algebraic Geometry Lecture Six: Grassmannians
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which can also be thought of as the number of paths from the lower-left corner to the upper-right corner of a k by n− k box, using unit steps up or right. insert Young diagram. The tangent space at pA is Hom(S,Q), where S is the vector space corresponding to pA, and Q = C /S. This has basis {ea ⊗ eb | a ∈ A, b 6∈ A} so the characters of the torus action are ta − tb, for a ∈ A and b 6∈ A. Given a ∈ A and b 6∈ A, set χ = ta−tb, and let A ′ = (Ar{a})∪{b}, so −χ = tb−ta is a character at pA′ . The corresponding curve Eχ,pA consists of points of the form
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Equivariant Cohomology in Algebraic Geometry Lecture Nine: Flag Varieties
Remark 0.1. We saw that the study of cohomology and equivariant cohomology of Grassmannians leads to interesting symmetric polynomials, namely, the Schur polynomials sλ(x) and sλ(x|t). These arise in contexts other than intersection theory and representation theory. For example, Griffiths asked which polynomials P in c1(E), . . . , cn(E) are positive whenever E is an ample vector bundle on an n...
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1.2. Schubert basis. To get more information, we must restrict to a torus. Take V = Cn, and let T be the subgroup of diagonal matrices in GLnC. We have the same description of H∗ TX, where X = Gr(k, n), but now Λ = ΛT = Z[t1, . . . , tn] and c(E) = ∏n i=1(1+ ti). Taking a T -invariant flag F•, we have T -invariant Schubert varieties Ωλ(F•). (In this section, we always assume a partition λ is co...
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σλ = |c T λi+j−i(Q− Fl+i−λi)|1≤i,j≤k. These determinants are variations of Schur polynomials, which we will call double Schur polynomials and denote sλ(x|y), where the two sets of variables are x = (x1, . . . , xk) and y = (y1, . . . , yn). (Here k ≤ n, and the length of λ is at most k.) Setting the y variables to 0, one recovers the ordinary Schur polynomials: sλ(x|0) = sλ(x). In fact, sλ(x|y)...
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تاریخ انتشار 2007