The Littlewood–Richardson rule states that the product of two Schur polynomials can be written as a finite weighted sum of Schur polynomials. More precisely $$ s_\lambda s_\mu = \sum_\nu c_{\lambda,\mu}^\nu s_\nu $$ where $c_{\lambda,\mu}^\nu$ is equal to the number of Littlewood–Richardson tableaux of skew shape $\nu /\lambda$ and of weight $\mu$.

Linked with some explicit computations in random matrix theory, I am hopping that a similar product rule may hold for the functions $\varphi_a(s)$ defined by $$ \varphi_a(x)=\frac{\det[e^{a_ix_j}]}{\prod_{i<j}(a_j-a_i)(x_j-x_i)},\qquad a=(a_1,\ldots,a_n),\quad x=(x_1,\ldots,x_n)\in\mathbb R^n. $$ More precisely, can we find a (discrete?) measure $\pi_{a,b}$ on $\mathbb R^n$ such that $$ \varphi_a(x)\varphi_b(x)=\int \varphi_c(x)\,\pi_{a,b}(\mathrm d c)\; ? $$ To give further context, if $A=\mathrm{diag}(a_1,\ldots,a_n)$ and $X=\mathrm{diag}(x_1,\ldots,x_n)$, then the Harish-Chandra-Itzykson-Zuber integral formula states that $$ \int_{U_n(\mathbb C)}e^{\mathrm{Tr}(AUXU^*)}\mathrm dU=c_n \varphi_a(x) $$ for some explicit constant $c_n$, where $\mathrm dU$ stands for the Haar measure on the unitary group $U_n(\mathbb C)$. These functions seem to play an analogue role as the $s_\lambda$'s in the representation of the semi-direct product of the Hermitian matrices with the unitary group via central automorphisms ($H\mapsto UHU^*$), although the latter correspondance is not clear to me either. If there is any reference available for this specific matter I would be also interested.