Given two sets of variables $X=\{x\_1,\cdots,x\_n\}$, $Y=\{y\_1,\cdots,y\_m\}$, and two partitions $\lambda$ and $\mu$. Is there a formula for the product of the Schur functions $s\_{\lambda}(X) s\_{\mu}(Y)$?


If $X=Y$ then the answer is given by the Littlewood–Richardson rule, but what if they are not the same? 


(I have also asked it in SE:
<https://math.stackexchange.com/questions/2307985/product-of-schur-functions>)



Product of Schur functions
If $X = (u\_1, u\_2, \ldots, u\_a, w\_1, w\_2, \ldots, w\_c)$ and $Y = (v\_1, v\_2, \ldots, v\_b, w\_1, \ldots, w\_c)$ with $u$'s, $v$'s and $w$'s disjoint, then 
$$s\_{\lambda}(u,w) s\_{\mu}(v,w) = \left( \sum\_{\alpha,\ \beta} c\_{\alpha \beta}^{\lambda} s\_{\alpha}(u) s\_{\beta}(w) \right) \left( \sum\_{\gamma,\ \delta} c\_{\gamma \delta}^{\mu} s\_{\gamma}(v) s\_{\delta}(w) \right)$$
$$=\sum\_{\alpha,\ \gamma} s\_{\alpha}(u) s\_{\gamma}(v) \sum\_{\beta,\ \delta} 
c\_{\alpha \beta}^{\lambda} c\_{\gamma \delta}^{\mu} s\_{\beta}(w) s\_{\delta}(w)$$
$$= \sum\_{\alpha,\ \gamma,\ \epsilon} s\_{\alpha}(u) s\_{\gamma}(v) s\_{\epsilon}(w) \sum\_{\beta,\ \delta} c\_{\alpha \beta}^{\lambda} c\_{\gamma \delta}^{\mu} c\_{\beta \delta}^{\epsilon}.$$


The first equality follows from $s\_{\lambda}(u,w) = \sum\_{\alpha} s\_{\alpha}(u) s\_{\lambda/\alpha}(w)$ (obvious from the definition using Young tableaux) and the second bullet point [here](https://en.wikipedia.org/wiki/Littlewood%E2%80%93Richardson_rule#Littlewood.E2.80.93Richardson_coefficients). The last equality uses the first bullet point at [the same link](https://en.wikipedia.org/wiki/Littlewood%E2%80%93Richardson_rule#Littlewood.E2.80.93Richardson_coefficients).


