# Littlewood–Richardson rule and the Harish-Chandra-Itzykson-Zuber integral

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.

Here is something that might be relevant for you. You can consider the integral: \begin{equation} C_{abc} = \prod_{ij} \left( \int_{-i\infty}^{i\infty} d X_{ij} \right) \varphi_a (X) \varphi_b(X) \varphi_c(X) \,, \end{equation} where I have replaced the eigenvalues of $X$ with the original Hermitian matrix. This "structure coefficient" can be written as \begin{equation} C_{abc} = C_0 \int dU dV dW \delta (UAU^* + VBV^* + WCW^*) \,, \end{equation} for some constant $C_0$, where $\delta$ is the Dirac delta function. This is the probability for three "random matrices" with eigenvalues $(a_i)$, $(b_i)$ and $(c_i)$ to add up to zero.

This probability has been computed by Tao and Knutson in:

http://www.ams.org/notices/200102/fea-knutson.pdf

This involves computing the volume of the moduli space of honeycomb diagrams. The honeycomb diagrams are diagrams made by connecting vertices emanating three rays forming an angle of $\pi/3$. There are finite length edges connecting the vertices, and three groups of rays, emanating to infinity in three directions at angles of $\pi/3$. The spacings of the rays in each direction encode the eigenvalues $(a_i)$, $(b_i)$ and $(c_i)$.

Tao and Knutson also tell you that you can also count the number of such diagrams when the spacings of the rays and the length of the edges are quantized. These reproduce the Littlewood-Richardson coefficients, the three representations encoded, again, in the quantized spacings of the emanating rays.

Thus $C_{abc}$ are literally a continuous version of the Littlewood-Richardson coefficient, which I think is what you are looking for.

• Thank you for your answer. I don't understand your first claim, can you elaborate/provide a reference? In particular I believe the Plancherel measure for anti-Hermitian matrices involves a squared Vandermonde of the eigenvalues. My question was indeed motivated by addition of matrices, and I agree that Knutson and Tao's works are good references. Notice also the volume you mention is highly non-explicit. But I think your answer goes in the good direction, hence my upvote. Aug 27, 2016 at 8:57