Puzzles and (equivariant) cohomology of Grassmannians

Puzzles and (equivariant) Cohomology of Grassmannians

The product of two Schubert cohomology classes on a Grassmannian Grk(C) has long been known to be a positive combination of other Schubert classes, and many manifestly positive formulae are now available for computing such a product (e.g., the Littlewood-Richardson rule or the more symmetric puzzle rule from A. Knutson, T. Tao, and C. Woodward [KTW]). Recently, W. Graham showed in [G], nonconst...

Bigraded Cohomology of Z/2-equivariant Grassmannians

Equivariant Cohomology in Algebraic Geometry Lecture Seven: Equivariant Cohomology of Grassmannians

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...

Equivariant Cohomology in Algebraic Geometry Lecture Eight: Equivariant Cohomology of Grassmannians Ii

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)...

Equivariant Cohomology in Algebraic Geometry Lecture Six: Grassmannians

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 ...