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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)

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    $\begingroup$ What kind of formula do you want? Generally if $f$ and $g$ are polynomials in distinct variables then the simplest form for their product is the obvious one! Maybe mathoverflow.net/questions/98494/… will be of interest. $\endgroup$ Jun 3 '17 at 12:27
  • $\begingroup$ The ideal answer would be a sum over certain symmetric functions, which specializes to the Littlewood–Richardson formula for $X=Y$ (I don't assume that $X$ and $Y$ are disjoint). $\endgroup$
    – Iman
    Jun 3 '17 at 12:34
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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. The last equality uses the first bullet point at the same link.

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