Littlewood-Richardson-Type Rule for Cohomology Ring of Grassmannians - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T18:49:54Z http://mathoverflow.net/feeds/question/4745 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/4745/littlewood-richardson-type-rule-for-cohomology-ring-of-grassmannians Littlewood-Richardson-Type Rule for Cohomology Ring of Grassmannians Dinakar Muthiah 2009-11-09T17:15:49Z 2009-11-09T17:19:04Z <p>The ordinary Grassmannian of k-planes in n-space is a coset space for \$GL_n\$. It is \$GL_n\$ mod a maximal parabolic. Here there is a nice basis given by Schubert varieties, which can be indexed by Young diagrams that fit in an (k)x(n-k) box. The structure constants for the cup product are then given by Littlewood-Richardson numbers.</p> <p>My question: is there a similarly nice picture for Grassmannians of arbitrary simple groups. Here the ordinary Grassmannian is replaced by \$G/P\$ where \$G\$ is a simple group and \$P\$ is a maximal parabolic. There are still Schubert varieties in this case, but I don't know how to say anything about the cup product.</p> http://mathoverflow.net/questions/4745/littlewood-richardson-type-rule-for-cohomology-ring-of-grassmannians/4747#4747 Answer by David Speyer for Littlewood-Richardson-Type Rule for Cohomology Ring of Grassmannians David Speyer 2009-11-09T17:19:04Z 2009-11-09T17:19:04Z <p>As yet, such a nice rule has only been formulated in the case that \$G/P\$ is minuscule or co-minuscule. See <a href="http://front.math.ucdavis.edu/0608.5276" rel="nofollow">Thomas and Yong</a> for details.</p>