Three responses to Allen Hatcher's question about Grassmanians, repeated below........DMD ________________________________________ Subject: Re: question abt Grassmanian Date: 11 Jul 2001 14:13:00 +0100 From: N.P.Strickland@sheffield.ac.uk > Consider the complex Grassmann manifold G_k,n of k-planes in C^n > (or take the real case if you prefer). Borel showed that the > integral cohomology ring of G_k,n is the polynomial ring on the > first k Chern classes of the canonical k-plane bundle E_k,n > modulo certain obvious relations coming from the Whitney sum formula > applied to the sum of E_k,n and its orthogonal (n - k)-plane > bundle. In spite of this simple description, the ring stucture is > well-known to be fairly complicated. My question is whether an > additive basis for the cohomology ring is formed by the monomials of > degree less than or equal to n - k in the first k Chern classes > of E_k,n. > This is true, and is proved in Lemma 6.9 and Remark 7.4 of my paper "Common subbundles and intersections of divisors" (available from http://www.shef.ac.uk/~pm1nps) for example. I'd be amazed if there wasn't an earlier reference, but I don't know of one. Neil Strickland ____________________________________ Subject: Re: question abt Grassmanian Date: Wed, 11 Jul 2001 10:00:28 -0500 From: Clarence Wilkerson 1) Is some of this material in Howard Hiller's book about the Schubert calculus? (late 80's?) 2) The proof that would come to my mind is that H^*(B(U(n) x U(m)) is a free H^*(BU(n+m)) module. Hence the EM spectral sequence converging to H^*(U(n+m)/(U(n)xU(m))) collapses. I would not be surprised if one could do this without the full EM SS also. 3) I think that in the general case of describing H^*(G/H), where H is a maximal rank subgroup that is the centralizer of some ( not neces. maixmal) torus, an additive basis of H^*(G/H) can be given in terms of the Weyl groups of G and H. Calculating the product structure wrt to this additive basis is related to the Schubert calculus. Clarence ___________________________________________________ Subject: Re: question abt Grassmanian Date: Wed, 11 Jul 2001 12:49:20 -0400 (Eastern Daylight Time) From: "Nicholas J. Kuhn" For some reason related to refereeing something, the question that Allen poses came up in a discussion I had with Bob Stong in early 1989. We ultimately came up with a one line proof, based on the "Giambelli formula", which appears on p112 of Howard Hiller's book Geometry of Coxeter Groups. The proof goes as follows... Firstly, H^*(Gr_k(C^{k+n}) = Z[c_1,...,c_k, d_1,...d_n]/cd, where the c_i are the chern classes of the canonical k-plane bundle, the d_i are the chern classes of the complementary n-plane bundle, and c and d are the total classes. Hatcher's question was whether the cohomology is additively spanned by words in the c_i of length at most n. By symmetry of course, we could try to show that the cohomology is additively spanned by words in the d_j of length at most k. As discussed in section III.3 of Hiller's book, there is a well known "Shubert cell" additive basis for the cohomology: there is a basis element for each nonincreasing k tuple of natural numbers between 0 and n. The Giambelli formula is a formula that writes each of these basis elements as the determinant of an explicit k x k matrix in the d_j's. Since k x k determinants are Z-linear combinations of words in the d_j of the required form, we see these words span. As Hatcher points out, a counting argument then shows they must be linearly independent. Summarizing... THM H^*(Gr_k(C^{k+n}) has an additive basis given by all mononials in the c_i of length at most n. By the way, Hiller's book, written in the early 80's before he was lured to Wall Street (ahead of his time!) is recommended. It has lots references to the topology, alg geometry, and combinatoric literature. Nick Kuhn