Schubert calculus

Remarks

For Grassmannians the constants $c_{\lambda \mu}^{\nu}$ are the Littlewood–Richardson coefficients, and they are nonnegative. The Pieri rule computes the product with $\sigma_k$, for $k = 1, \ldots, n - d$. It says that $\sigma_k \sigma_\lambda$ is the sum of $\sigma_\nu$ over the $\nu$ obtained from $\lambda$ by adding $k$ boxes, no two in the same column. Only the $\nu$ that fit in the $d \times (n - d)$ rectangle are counted.

Example: in $\mathrm{Gr}(2, 4)$, the Pieri rule gives $\sigma_1^2 = \sigma_2 + \sigma_{1,1}$, $\sigma_1 \sigma_2 = \sigma_1 \sigma_{1,1} = \sigma_{2,1}$ and $\sigma_1 \sigma_{2,1} = \sigma_{2,2}$. So $\sigma_1^4 = 2 \sigma_{2,2}$, and $\sigma_{2,2}$ is the class of a point. Hence $2$ lines in $\mathbb{P}^3$ meet four general lines.

Non-example: the count of lines meeting four lines in a common plane is not given by $\sigma_1^4$. The four Schubert varieties are not in general position, and they meet in a positive-dimensional set.