Littlewood–Richardson rule

For induction of $S^{\mu }⊠S^{\lambda }$ to the symmetric group on $|\mu |+|\lambda |$ letters, the Specht filtration multiplicity of $S^{\nu }$ is the number of Semistandard Young tableaux of skew shape $\nu /\mu$, content $\lambda$, and lattice word read from right to left in each row, taking rows from top to bottom. In characteristic zero these are irreducible constituents; in positive characteristic they remain Specht factors rather than simple constituents. Content $(1)$ gives the add-one-cell induction rule.

The nonnegative integer $c_{\alpha \beta }^{\nu }$ is the multiplicity of $S^{\nu }$ in ${\mathrm{Ind}}_{S_{|\alpha |}\times S_{|\beta |}}^{S_{|\nu |}}(S^{\alpha }⊠S^{\beta })$ in characteristic zero. The Littlewood–Richardson rule counts it by Semistandard Young tableaux of skew shape $\nu /\alpha$, content $\beta$, and lattice word in right-to-left, top-to-bottom reading order. For example $c_{(3,2,1),(3,2)}^{(5,3,2,1)}=3$: the two new top cells must be ones, and the remaining reading letters can be $122,212,221$. This coefficient also multiplies Schur functions in the induction ring of symmetric-group characters.

In the induction ring of ordinary symmetric-group characters, $[\lambda ][r]$ is the sum, with multiplicity one, of $[\nu ]$ for which $\nu /\lambda$ is a horizontal strip of $r$ cells. Here $[r]$ is the trivial representation of $S_r$. In the Littlewood–Richardson rule, content $(r)$ means every skew entry is $1$; strict column increase is possible exactly when there is at most one new cell in every column. This proves the rule.