Reference for exponential Vandermonde determinant identity

Question

I recently stumbled upon the following identity, valid for any real numbers $\alpha_1,\dots,\alpha_n$ and $\lambda_{n1} \leq \dots \leq \lambda_{nn}$:

$$ \mathrm{det}( e^{\alpha_i \lambda_{nj}} )_{1 \leq i,j \leq n} = V(\alpha) \int_{GT_\lambda} \exp( \sum_{i=1}^n \sum_{j=1}^i \lambda_{ij} (\alpha_{n+1-i}-\alpha_{n-i}))$$ where $V(\alpha)$ is the Vandermonde determinant $$ V(\alpha) := \prod_{1 \leq i < j \leq n} (\alpha_j - \alpha_i),$$ $GT_\lambda$ is the Gelfand-Tsetlin polytope of tuples $(\lambda_{ij})_{1 \leq j \leq i < n}$ obeying the interlacing relations $\lambda_{i+1,j} \leq \lambda_{i,j} \leq \lambda_{i,j+1}$ and with the usual Lebesgue measure, and one has the convention $\alpha_0 := 0$. Thus for instance when $n=1$ one has $$ e^{\alpha_1 \lambda_{11}} = \exp( \lambda_{11} \alpha_1 )$$ when $n=2$ one has $$ \mathrm{det} \begin{pmatrix} e^{\alpha_1 \lambda_{21}} & e^{\alpha_1 \lambda_{22}} \\ e^{\alpha_2 \lambda_{21}} & e^{\alpha_2 \lambda_{22}} \end{pmatrix} $$ $$= (\alpha_2 - \alpha_1) \int_{\lambda_{21} \leq \lambda_{11} \leq \lambda_{22}} \exp( \lambda_{11} (\alpha_2-\alpha_1) + \lambda_{21} \alpha_1 + \lambda_{22} \alpha_1 )\ d\lambda_{11}$$ and when $n=3$ one has $$ \mathrm{det} \begin{pmatrix} e^{\alpha_1 \lambda_{31}} & e^{\alpha_1 \lambda_{32}} & e^{\alpha_1 \lambda_{33}} \\ e^{\alpha_2 \lambda_{31}} & e^{\alpha_2 \lambda_{32}} & e^{\alpha_2 \lambda_{33}} \\ e^{\alpha_3 \lambda_{31}} & e^{\alpha_3 \lambda_{32}} & e^{\alpha_3 \lambda_{33}} \end{pmatrix} $$ $$ = (\alpha_2 - \alpha_1) (\alpha_3 - \alpha_1) (\alpha_3 - \alpha_2) \int_{\lambda_{31} \leq \lambda_{21} \leq \lambda_{32}} \int_{\lambda_{32} \leq \lambda_{22} \leq \lambda_{33}} \int_{\lambda_{21} \leq \lambda_{11} \leq \lambda_{22}}$$ $$ \exp( \lambda_{11} (\alpha_3-\alpha_2) + \lambda_{21} (\alpha_2-\alpha_1) + \lambda_{22} (\alpha_2-\alpha_1) + \lambda_{31} \alpha_1 + \lambda_{32} \alpha_1 + \lambda_{33} \alpha_1)$$ $$ d \lambda_{11} d\lambda_{22} d\lambda_{21},$$ and so forth.

The identity can be proven easily by induction. I first discovered it by starting with the Schur polynomial identity $$ \mathrm{det}( x_j^{a_i} )_{1 \leq i,j \leq n} = V(x) \sum_T x^{|T|}$$ where $0 \leq a_1 < \dots < a_n$ are natural numbers in increasing order, $T$ ranges over column-strict Young tableaux of shape $a_n-n+1, \dots, a_2-1, a_1$ with entries in $1,\dots,n$, and $x^{|T|} := x_1^{c_1} \dots x_n^{c_n}$ where $c_i$ is the number of occurrences of $i$ in $T$, and taking a suitable "continuum limit" as the $a_i$ go to infinity and the $x_j$ go to one in a particular fashion. It can also be derived from the Duistermaat-Heckmann formula for the Fourier transform of Schur-Horn measure, combined with the fact that this measure is the pushforward of Lebesgue measure on the Gelfand-Tsetlin polytope under a certain linear map.

Note that the identity also provides an immediate proof that any $n$ distinct exponential functions on $n$ distinct real numbers are linearly independent.

I am certain that this formula already appears in the literature, and perhaps even has a standard name, but I was unable to locate it with standard searches. So my question here is if anyone recognizes the formula and can supply a reference for it.

Answer 1

This looks like a special case of a formula by Samson Shatashvili related to the HCIZ integral as mentioned in Ryan's answer. Compare, in particular the two ways of computing $\langle 1\rangle$ given by Equations 3.2 and 3.4 in "Correlation Functions in The Itzykson-Zuber Model" (thanks to Leonid Petrov for letting me know about this reference in his answer to this MO question).

Answer 2

Write $\beta = \lambda_n$, the top row of your GT patterns. It's a theorem of [Baryshnikov] that if we choose a uniformly random point in the polytope GT${}_\lambda$, it's equivalent to choosing a Haar-random Hermitian matrix with spectrum $\beta$ and then taking its "principal minors". (I've also seen this fact credited to Weyl, and others.) More precisely, let $B = \mathrm{diag}(\beta)$, and form a matrix $X = U B U^\dagger$, where $U$ is a random unitary. Then let $\lambda_{11}$ be the top-left entry of $X$, let $\lambda_{21}, \lambda_{22}$ be the eigenvalues of the top-left $2 \times 2$ submatrix of $X$, ..., and let $\lambda_{n1}, \dots, \lambda_{nn}$ be the eigenvalues of the top-left $n \times n $ submatrix of $X$ (namely, $\beta$). Then $\lambda$ is uniformly random in the polytope GT${}_\lambda$.

This probability distribution on $\lambda$ is basically your integral, but we have to divide by the volume of the polytope, which is $V(\lambda)/[(n-1)! (n-2)! \cdots 2! 1!]$. I guess this is standard? If not, it's also in Baryshnikov.

Having done so, your identity is the Harish-Chandra--Itzykson--Zuber identity applied to the matrices $A = \mathrm{diag}(\alpha)$ and $B$. This follows by inferring the diagonal entries of $X$ from the Gelfand-Tsetlin pattern $\lambda$, which you can do because the Gelfand--Tsetlin pattern gives you the traces of all the top-left submatrices.

(By the way, I think the [Faraut] paper referenced below has a good exposition of some related things.)

Baryshnikov, Yu., GUEs and queues, Probab. Theory Relat. Fields 119, No.2, 256-274 (2001). ZBL0980.60042.

Faraut, Jacques, Rayleigh theorem, projection of orbital measures and spline functions, Adv. Pure Appl. Math. 6, No. 4, 261-283 (2015). ZBL1326.15058.