Littlewood-Richardson number

Definition
The Littlewood-Richardson number is a number associated with three unordered integer partitions $$\mu,\nu,\lambda$$, and is denoted $$c^\lambda_{\mu,\nu}$$. 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, $$c^\lambda_{\mu,\nu}$$ 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 $$\nu$$.