The Schubert Calculus, Braid Relations, and Generalized Cohomology

Abstract

Abstract.Let X be the flag variety of a compact Lie group and let h* be a complex-oriented generalized cohomology theory. We introduce operators on h*(X) which generalize operators introduced by Bernstein, Gel fand, and Gel fand for rational cohomology and by Demazure for if-theory. Using the Becker-Gottlieb transfer, we give a formula for these operators, which enables us to prove that they satisfy braid relations only for the two classical cases, thereby giving a topological interpretation of a theorem proved by the authors and extended by Gutkin.One of the central issues in Lie theory is the geometry of the flag variety associated to a compact Lie group G. An important problem concerning the flag variety is the Schubert calculus, which studies the ring structure of the cohomology of the flag variety. Work initiated by Borel, Bott and Kostant, which culminated in a paper by Bernstein, Gel'fand and Gel'fand [BGG], gave a complete solution to the problem. Demazure studied the same problem for ./sT-theory. Moreover, these techniques have been generalized to the Kac-Moody situation by Kac-Peterson, Kostant-Kumar, and others. This work has focussed on algebro-geometric properties of the flag variety.Here, on the other hand we study the flag variety from the point of view of algebraic topology. As a consequence, not only do we recover the classical results described above, but we extend these results to a certain class of cohomology theories-those which are complex-oriented. Examples of complex-oriented theories include ordinary cohomology, A^-theory, complex cobordism, and elliptic cohomology. Since the context we have chosen in very general, the proofs are universal and are often simpler than the classical arguments.In the work of BGG, a crucial role is played by operators Ai associated to each simple reflection s¡ of the Weyl group of G (defined by Demazure in if-theory). These operators Ai satisfy the braid relations, which are the relations between pairs of simple reflections. In this paper, we generalize the Ai to give operators Di acting on h*(G/T) for any complex-oriented theory h*. We prove that braid relations are satisfied only for cohomology theories with the formal group law of rational cohomology or of AMheory (Theorem

Segre classes and Damon–Kempf–Laksov formula in algebraic cobordism

“…In order to better appreciate the significance of our formula, it may be worth placing it within the wider framework of generalised Schubert calculus. In recent years a lot of effort has been devoted to lift results of classical Schubert calculus to Ω * , in a fashion similar to what Bressler-Evens did in [6,7] for topological cobordism. In particular, this line of research was pioneered by Calmés-Petrov-Zanoulline ( [8]) and ) who studied the algebraic cobordism of flag manifolds.…”

Motivic decompositions of twisted flag varieties and representations of Hecke-type algebras

“…If E is a generic G-torsor, then D F can be replaced by the formal affine Demazure algebra D F . The theory of such algebras and formal push-pull operators has been recently developed in [6], [19], [7], [8], [9] motivated by Bernstein-Gelfand-Gelfand [1], Demazure [11], [12], Bressler-Evens [2], [3], Kostant-Kumar [22], [21], Brion [4], Totaro [29] and Edidin-Graham [13]. The key properties of D F are -It is a free module over the T -equivariant oriented cohomology ring S = h T (k) of a point, where T is a split maximal torus in G [7].…”

Smooth Schubert varieties and generalized Schubert polynomials in algebraic cobordism of Grassmannians

“…In this subsection, we present explicit results concerning elliptic cohomology, i.e for the hyperbolic FGL, of Gr (2,4). We compute the polynomial representing any Bott-Samelson class as well as their products.…”

“…Let X = Gr (2,4) and let λ be a partition. Denote by L λ the polynomial in Ω * (G/B) ≃ L[x 1 , x 2 , x 3 , x 4 ]/S representing the pull-back in G/B of the cobordism class [ X λ → X] where X λ is the Bott-Samelson resolution of X λ .…”

“…In particular, we show that some of the smooth Schubert varieties satisfy a certain symmetry, see Corollary 4.2. For generalized Schubert polynomials associated to other cells, this is no longer true already when looking at Gr (2,4), see Proposition 4.5.…”