# Writing down cohomology groups of the complex Grassmannian

I am studying a homogeneous space and would like to know its cohomology groups. Using some sequences and fibrations I have figured out some of these groups, but largely in terms of the cohomology group of another space. That space is the complex Grassmannian:

$SU(m+n)/SU(m)\times SU(n)$

The cohomology ring of the Grassmannian is complicated. The treatment in http://www-personal.umich.edu/~jblasiak/grassmannian.pdf is understandable and I am working through it. However, I am a little bit confused about how to actually recover the "classical" cohomology groups from something like example 5.4, page 15 in the pdf. I find myself in the strange position of being told that these cohomology groups are known and, although complicated, possible to write down, without seeing how.

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What don't you understand about example 5.4? – Qiaochu Yuan Mar 23 '12 at 0:53
Don't forget that the cohomology groups of any space go together to form the cohomology ring whose product is the cup product. I guess the idea is that by doing these Young tableux calculations, you're really working out the cup-product structure of the cohomology ring. – you Mar 23 '12 at 2:29