Consider the cohomology ring of the Grassmannian of k-planes in complex n-space. It has a standard presentation as a quotient of the ring of symmetric functions. In this presentation, the Schur functions are mapped to the Schubert classes, thus have a nice geometric interpretation.

One can generalise the Schur functions to skew-Schur functions. Do these also have a nice geometric interpretation?

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    $\begingroup$ I don’t know for sure but I think they should correspond to Richardson varieities the same way ordinary Schur functions correspond to Schubert varieties. $\endgroup$ – Sam Hopkins Nov 27 '17 at 18:58

This is discussed in Stanley's paper Some combinatorial aspects of the Schubert calculus. Corollary 3.7 says that under the natural isomorphism given by the Borel presentation of $H^*(G/P)$ which sends an ordinary Schur function $s_{\lambda}$ to the class of the Schubert variety $X_{\lambda}$, a skew Schur function $s_{\lambda / \mu}$ is sent to the class of the Richardson variety $X_{\lambda}^{\mu}$.

Note that Stanley calls Richardson varieties skew Schubert varieties in this paper. Unfortunately, the version of the paper that is online at Stanley's website has some printing defects that make some of the pages illegible.


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