Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I been told that there exists an integration formula, which states (or something of this sort) $$ \int_{U(N)} dU \det[(\mathbb I+XUYU^{-1})^{-r}\propto \frac{\det(1+x_iy_j)^{N-r-1}]_{i,j}}{\Delta_N(x)\Delta_N(y)}, $$ where $dU$ is the Haar-measure, $X=\text{diag}(x_1,\ldots,x_N)$ is a positive definite diagonal matrix with $x_i\neq x_j$ for $i\neq j$ (similar for $Y$), and $$ \Delta_N(x)=\prod_{1\leq i<j\leq N}(x_i-x_j) $$ is a Vandermonde determinant.

This is somewhat similar to a Harish-Chandra-Itzykson-Zuber integral, but I haven't seen this specific kind before. Are anyone familiar with this kind of integrals, and how to solve them? I would be particular interested in references to the literature.

share|cite|improve this question
up vote 1 down vote accepted

Since I found a reference which answers this question (and noone has provided any other solution) I will give a short answer to my own question.

This type of integrals has been studied in the context of two-matrix models. The integral in question has been discussed in arXiv:0804.0873 (section 3), while a larger class of integrals has been discussed in arXiv:0512056 [math-ph] (appendix A).

share|cite|improve this answer

This statement does not seem to be correct, because if X tends to the zero matrix, the left hand side is well defined but the right hand side is not (notice that HCIZ integral does not suffer from this).

share|cite|improve this answer
Ups! I had misunderstood the right hand side. I read it as if the numerator had the determinant of the identity plus something... Sorry – Marcel Mar 24 '13 at 23:11

This identity, and others of similar type, are proved explicitly in: J. Harnad and A. Yu. Orlov, ``Fermionic construction of partition functions for two-matrix models and perturbative Schur function expansions'', J. Phys. A, 39, 8783--8809 (2006). (Appendix A]

share|cite|improve this answer
Thank you very much for your comment. I presume that this type of integrals dates back to Gross and Richards ( Trans. Amer. Math. Soc. 301 (1987), 781 , J. Approx. Theory 59 (1989) 224 , Bulletin of AMS 24 (1991) 349 ), which, more recently, have been extended by Orlov and yourself? – Jesper Ipsen Jul 7 '14 at 11:43

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.