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  "body": "\n## Definition\n\nLet $1 \\le d \u003c n$, and let $\\mathrm{Gr}(d, n)$ be the complex [[Grassmannian|CYPA882RB9]].\nFor each partition $\\lambda$ in the $d \\times (n - d)$ rectangle, let $\\sigma_\\lambda \\in H^{2 \\abs{\\lambda}}(\\mathrm{Gr}(d, n); \\Z)$ be the class of the [[Schubert variety|Z014Q8KPDD]] $X_\\lambda$.\nThese classes form a $\\Z$-basis of the [[cohomology ring|RDT4785M4D]] $H^{*}(\\mathrm{Gr}(d, n); \\Z)$.\n*Schubert calculus* is the computation of the structure constants $c_{\\lambda \\mu}^{\\nu}$ in\n$$\\sigma_\\lambda \\cdot \\sigma_\\mu = \\sum_\\nu c_{\\lambda \\mu}^{\\nu} \\, \\sigma_\\nu,$$\nand their use to count intersection points of Schubert varieties in general position.\nThe same name is used for the analogous computations in the [[Chow ring|EBEF6PEK4N]] and for flag varieties.\n\n## Remarks\n\nFor Grassmannians the constants $c_{\\lambda \\mu}^{\\nu}$ are the Littlewood–Richardson coefficients, and they are nonnegative.\nThe Pieri rule computes the product with $\\sigma_k$, for $k = 1, \\ldots, n - d$.\nIt 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.\nOnly the $\\nu$ that fit in the $d \\times (n - d)$ rectangle are counted.\n\nExample: 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 $\\sigma_1 \\sigma_{2,1} = \\sigma_{2,2}$.\nSo $\\sigma_1^4 = 2 \\sigma_{2,2}$, and $\\sigma_{2,2}$ is the class of a point.\nHence $2$ lines in $\\mathbb{P}^3$ meet four general lines.\n\nNon-example: the count of lines meeting four lines in a common plane is not given by $\\sigma_1^4$.\nThe four Schubert varieties are not in general position, and they meet in a positive-dimensional set.\n",
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