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  "body": "\n## Statement\n\nLet $\\lambda$ and $\\mu$ be [[partitions|0JV30XN73Z]], and let $s_\\lambda$ denote the [[Schur polynomial|43VFMJCDYJ]] (in any number of variables, or the Schur function).\nThen\n$$s_\\lambda \\, s_\\mu = \\sum_{\\nu} c^{\\nu}_{\\lambda \\mu} \\, s_\\nu,$$\nwhere $c^{\\nu}_{\\lambda\\mu}$ is the number of Littlewood–Richardson tableaux of shape $\\nu / \\lambda$ and content $\\mu$.\nSuch a tableau is a filling $T$ of the skew shape $\\nu / \\lambda$ with these properties.\nHere $\\nu / \\lambda$ is the set of cells of the [[Young diagram|096AHMB3P4]] of $\\nu$ that are not in the diagram of $\\lambda$.\n\n1. The entries weakly increase along rows and strictly increase down columns.\n2. For each $k$, exactly $\\mu_k$ entries are equal to $k$.\n3. Read the rows from right to left, from the top row down. In every initial segment of this word, for each $k \\ge 1$, the entry $k$ occurs at least as often as $k+1$.\n\n## Remarks\n\nIn particular $c^{\\nu}_{\\lambda\\mu} = 0$ unless the diagram of $\\lambda$ lies in the diagram of $\\nu$ and $\\abs{\\nu} = \\abs{\\lambda} + \\abs{\\mu}$.\nThe coefficients are symmetric: $c^{\\nu}_{\\lambda\\mu} = c^{\\nu}_{\\mu\\lambda}$.\nModern proofs use the [[Robinson–Schensted–Knuth correspondence|8JW8F4VVED]] or Schützenberger's jeu de taquin, which turns each tableau of shape $\\nu/\\lambda$ into a straight-shape [[Young tableau|Q6TH9PH2R7]].\n\nExample: take $\\lambda = (1)$ and $\\mu = (1, 1)$.\nFor $\\nu = (2, 1)$ the two cells of $\\nu / \\lambda$ are in different rows and columns.\nThe reading word must start with $1$, so the only tableau has $1$ in the first row and $2$ in the second row.\nFor $\\nu = (1, 1, 1)$ the two cells form a column filled with $1$ above $2$, and the reading word $1\\,2$ is allowed.\nFor $\\nu = (3)$ the cells form a row $1\\,2$ whose reading word $2\\,1$ fails clause 3.\nHence $s_{(1)} s_{(1,1)} = s_{(2,1)} + s_{(1,1,1)}$.\n\n## Explanations\n\n### L1 human\n\nSchur polynomials are building blocks, and the product of two of them is again a sum of them.\nThe rule says how many times each building block appears: count certain fillings of a skew shape with numbers.\nThe reading condition makes sure that, at every point while reading, the $1$s are ahead of the $2$s, the $2$s ahead of the $3$s, and so on.\n\n### L2 human\n\nThe numbers $c^\\nu_{\\lambda\\mu}$ have several meanings at once.\nThey are the multiplicities in the tensor product of two irreducible polynomial representations of the general linear group.\nThey are the multiplicities of the irreducible representations of $S_{\\abs\\nu}$ in the representation induced from the irreducible representation of $S_{\\abs\\lambda} \\times S_{\\abs\\mu}$ indexed by the pair $(\\lambda, \\mu)$.\nThey are also the structure constants of the cohomology ring of a Grassmannian in the basis of Schubert classes.\nWhen $\\mu = (k)$ is a single row, the rule reduces to Pieri's rule.\nIt says that $s_\\lambda s_{(k)}$ is the sum of $s_\\nu$ over the $\\nu$ obtained by adding $k$ cells to $\\lambda$, no two in the same column.\n",
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