Littlewood–Richardson rule

In mathematics, the Littlewood–Richardson rule is a combinatorial description of the coefficients that arise when decomposing a product of two Schur functions as a linear combination of other Schur functions. These coefficients are natural numbers, which the Littlewood–Richardson rule describes as counting certain skew tableaux. They occur in many other mathematical contexts, for instance as multiplicity in the decomposition of tensor products of finite-dimensional representations of general linear groups, or in the decomposition of certain induced representations in the representation theory of the symmetric group, or in the area of algebraic combinatorics dealing with Young tableaux and symmetric polynomials.

Littlewood–Richardson coefficients depend on three partitions, say ${\displaystyle \lambda ,\mu ,\nu }$, of which ${\displaystyle \lambda }$ and ${\displaystyle \mu }$ describe the Schur functions being multiplied, and ${\displaystyle \nu }$ gives the Schur function of which this is the coefficient in the linear combination; in other words they are the coefficients ${\displaystyle c_{\lambda ,\mu }^{\nu }}$ such that

${\displaystyle s_{\lambda }s_{\mu }=\sum _{\nu }c_{\lambda ,\mu }^{\nu }s_{\nu }.}$

The Littlewood–Richardson rule states that ${\displaystyle c_{\lambda ,\mu }^{\nu }}$ is equal to the number of Littlewood–Richardson tableaux of skew shape ${\displaystyle \nu /\lambda }$ and of weight ${\displaystyle \mu }$.