Littlewood–Richardson rule

Remarks

In particular $c^{\nu}_{\lambda\mu} = 0$ unless the diagram of $\lambda$ lies in the diagram of $\nu$ and $\abs{\nu} = \abs{\lambda} + \abs{\mu}$. The coefficients are symmetric: $c^{\nu}_{\lambda\mu} = c^{\nu}_{\mu\lambda}$. Modern proofs use the Robinson–Schensted–Knuth correspondence or Schützenberger's jeu de taquin, which turns each tableau of shape $\nu/\lambda$ into a straight-shape Young tableau.

Example: take $\lambda = (1)$ and $\mu = (1, 1)$. For $\nu = (2, 1)$ the two cells of $\nu / \lambda$ are in different rows and columns. The reading word must start with $1$, so the only tableau has $1$ in the first row and $2$ in the second row. For $\nu = (1, 1, 1)$ the two cells form a column filled with $1$ above $2$, and the reading word $1\,2$ is allowed. For $\nu = (3)$ the cells form a row $1\,2$ whose reading word $2\,1$ fails clause 3. Hence $s_{(1)} s_{(1,1)} = s_{(2,1)} + s_{(1,1,1)}$.

Going deeper

For a student

The numbers $c^\nu_{\lambda\mu}$ have several meanings at once. They are the multiplicities in the tensor product of two irreducible polynomial representations of the general linear group. They 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 $(\lambda, \mu)$. They are also the structure constants of the cohomology ring of a Grassmannian in the basis of Schubert classes. When $\mu = (k)$ is a single row, the rule reduces to Pieri's rule. It 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.