Abusive dimensions and cohomology of Grassmannians

Yesterday I complained about the abuse of the word "dimension" by a lot of popular authors, namely the fact that they tried to order the dimensions.

Being a discrete mathematician, I don't think about dimensions all that often. But I'm taking a course in algebraic topology (it's required for my program); I have a habit of trying to twist the problems there into combinatorics problems whenever possible. Recently we were asked to compute the cohomology of the Grassmannian manifold G_nR^k. This Grassmannian is just the manifold of n-dimensional subspaces of k-dimensional space passing through the origin. It is of dimension k(n-k), as we can see by "dimension-counting". To obtain such a subspace we pick n orthogonal unit vectors in k-space. The first one is chosen from S^k-1 and thus we have k-1 "degrees of freedom", contributing k-1 to the dimension. The second one is chosen from a (k-2)-sphere in the (k-1)-dimensional orthogonal complement of this first vector; this contributes k-2 dimensions. And so on, down to k-n dimensions for the nth vector. But we've overcounted by 1 + 2 + ... + n-1, since we've actually chosen an n-dimensional flag in k dimensions; thus we need to quotient out by the orthogonal group SO(n), which has this dimension. Thus the dimension of the Grassmanian is

[(k-1) + (k-2) + ... + (k-n)] - [0 + 1 + ... + (n-1)] = n(k-n).

As you may have noticed, I used "dimension" in two senses there; the rigorous sense of the topological invariant which is always an integer, and the non-rigorous sense of "degree of freedom". At this point my complaint from yesterday is still valid, because I never ordered the dimensions, and that was the cardinal sin I attributed to people who don't know what they're talking about.

But then I caught myself saying things like "the cohomology of G₂R⁵ in the second dimension is Z²." That's no good.

As it turns out, I rediscovered the fact that the dimension of H^m(G_nR^k) is the number of integer partitions of m into at most k-n parts, none of which exceed n; in the case I just mentiond we have m = 2, n = 2, k = 5, so we're looking for integer partitions of 2 into 2 parts which don't exceed 3. There are two of them, namely 2 and 1 + 1. This isn't too interesting, but say we want to know, for example, the dimension of H₉(G₃R¹⁰)? Then we can just write down all the partitions of 9 into three parts which are at most 7;

7+2+0, 7+1+1, 6+3+0, 6+2+1, 5+4+0, 5+3+1, 5+2+2, 4+4+1, 4+3+2, 3+3+3

and there are ten of them; thus H₉(G₃R¹⁰) = Z¹⁰. You can write down generating functions for these sequences of integers; I suppose one could even ask about the asymptotic behavior of these dimensions, although I don't think a topologist would care. (I can't find a source for this fact, but I'm actually pretty sure I've heard it before; the two obvious sources to me were Hatcher's Algebraic Topology and Stanley's Enumerative Combinatorics, which are both books I know reasonably well, and I found the place in each where this should be mentioned. Stanley just says that there is a connection to the cohomology of the Grassmannian and leaves it at that; Hatcher talks about symmetric polynomials but never sinks to finding actual numbers.)

But as I said, I caught myself saying things like "what's the second-dimensional cohomology of G₂R⁵"? Second-dimensional. And I've heard this usage from Real Algebraic Topologists, too. So apparently what was bothering me yesterday was something more subtle than this abuse of language.