# Reference for exponential Vandermonde determinant identity

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.

• Maybe related : diva-portal.org/smash/get/diva2:1073960/FULLTEXT02.pdf Jul 31, 2017 at 17:45
• It looks close to the Harish-Chandra--Itzykson-Zuber identity, using the fact (Baryshnikov) that picking a random point in the GT${}_\lambda$ polytope is equivalent to picking a Haar-random Hermitian matrix with spectrum $\lambda_n$ and then taking its "principal minors" -- i.e., the spectra of its top-left square submatrices... Jul 31, 2017 at 17:48

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).

• Pity I can't accept both answers. I've accepted this one as it is an earlier reference that explicitly states two identities which, when combined, give the identity stated in the post. Aug 1, 2017 at 17:33
• @TerryTao: Indeed, it's too bad MO does not allow multiple accepted answers (as far as I know). BTW the paper by Shatashvili mentions earlier work with Alexeev and Faddeev but it's behind a paywall and I don't have access. It could be that your identity is there too. Aug 1, 2017 at 17:53

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.

• I can't believe I had forgotten about the HCIZ formula, since I actually blogged about it: terrytao.wordpress.com/2013/02/08/… . I'm pretty sure though that the observation that the spectra of minors of a random element of a U(n) coadjoint orbit are distributed uniformly within the GT polytope is much older than Baryshnikov, though. Jul 31, 2017 at 18:41
• Quite possible, yes, though people do often cite Baryshnikov. Olshanksi discusses it a bit briefly here -- arxiv.org/pdf/1302.7116.pdf -- and says it appears 'hidden' in Gelfand-Naimark'57. This paper of O'Connell -- arxiv.org/abs/1201.4849 -- calls it "well-known", but cites Baryshnikov. Jul 31, 2017 at 18:55