Littlewood-Richardson number

Definition

The Littlewood-Richardson number is a number associated with three unordered integer partitions $\mu ,\nu ,\lambda$, and is denoted $c_{\mu ,\nu }^{\lambda }$. It is nonzero only in some cases where the size of the number partitioned by $\lambda$ is the sum of the numbers partitioned by $\mu$ and $\nu$. We assume that we are working with countably many variables.

The Littlewood-Richardson numbers can be defined in a number of ways:

1. They are the structure constants for the Schur polynomials, viewed as a basis for the space of symmetric polynomials on countably many variables.
2. They are the structure constants for the subring generated by the Schur elements in the tableau ring.
3. Suppose $l,m,n$ are the numbers partitioned by $\lambda ,\mu \,nu$ respectively where $l=m+n$. Then, $c_{\mu ,\nu }^{\lambda }$ is the multiplicity of the irreducible representation corresponding to $\lambda$ in the induced representation from the Young subgroup $S_{m}\times S_{n}$ to $S_{l}$ of the outer tensor product of linear representations corresponding to $\mu$ and $\nu$.