# Is there a geometric interpretation of skew Schur functions?

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?

• 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. 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 $X_{\lambda}^{\mu}$.