Review: Schubert calculus

There is a strong connection between Schubert calculus and the computation of spectra. We review here the Grassmann variety, Schubert cells, Schubert subvarieties and then Schubert cycles and how to compute their product. We then review vector bundles over the sphere, clutching functions and the universal bundle. We discuss the Chern characteristic classes and their connection to Schubert cycles; indeed they generate the same cohomology ring. 1. Grassmannian This is the more formal review from Ledoux, Malham and Thümmler [11]. It has been been adapted slightly to conform with the more traditional exposition in the literature, which is to use row-span rather than column-span for the k-planes. Our main references are Chern [2], Griffiths and Harris [6], Milnor and Stasheff [12], Montgomery [13], Steenrod [17] and Warner [18, p. 130]. 1.1. Stiefel and Grassmann manifolds. A k-frame is a k-tuple of k 6 n linearly independent vectors in C. The Stiefel manifold V(n, k) of k-frames is the open subset of C of all k-frames centred at the origin. The set of k-dimensional subspaces of C forms a complex manifold Gr(n, k) called the Grassmann manifold of k-planes in C (see Steenrod [17, p. 35] or Griffiths and Harris [6, p. 193]). The fibre bundle π : V(n, k) → Gr(n, k) is a principal fibre bundle. For each y in the base space Gr(n, k), the inverse image π(y) is homeomorphic to the fibre space GL(k) which is a Lie group; see Montgomery [13, p. 151]. The projection map π is the natural quotient map sending each k-frame centered at the origin to the k-plane it spans; see Milnor and Stasheff [12, p. 56]. 1.2. Representation. Following the exposition in Griffiths and Harris [6], any k-plane in C can be represented by an k × n matrix of rank k, say Y ∈ C. Any two such matrices Y and Y ′ represent the same k-plane element of Gr(n, k) if and only if Y ′ = uY for some u ∈ GL(k) (the k-dimensional subspace elements are invariant to rank k closed transformations mapping k-planes to k-planes). Let j = {i1, . . . , ik} ⊂ {1, . . . , n} denote a multi-index of cardinality k. Let Yj◦ ⊂ C denote the (n− k)-plane in C spanned by the vectors {ej : j 6∈ j} and Uj = { Y ∈ Gr(n, k) : Y ∩ Yj◦ = {0} } . In other words, Uj is the set of k-planes Y ∈ Gr(n, k) such that the k×k submatrix of one, and hence any, matrix representation of Y is nonsingular (representing a coordinate patch labelled by j). 1