Before you jump on the "duplicate" buttom, let me say that I do not want to hear about Weingarten calculus and I do not want to see a character of the symmetric group.

What I would like is a formula for the (normalized) Haar measure integral $$ \int_{U(N)} g_{i_1 j_1}\cdots g_{i_n j_n} {\bar{g}}_{k_1 l_1}\cdots {\bar{g}}_{k_n l_n}\ d\mu(g) $$ of the form $$ \left.\mathcal{D}\ g_{i_1 j_1}\cdots g_{i_n j_n} {\bar{g}}_{k_1 l_1}\cdots {\bar{g}}_{k_n l_n}\right|_{g=\bar{g}=0} $$ where $\mathcal{D}$ is an explicit constant coefficient differential operator of infinite order. Of course, in this formula the $g$'s and $\bar{g}$'s are treated are $2N^2$ completely unrelated formal variables.

As an example of what I would like, in the case of $SU(N)$ and $\bar{g}$-free monomials $$ \mathcal{D}=\sum_{n=0}^{\infty} \frac{0!1!\cdots (N-1)!}{n!(n+1)!\cdots(n+N-1)!}\ ({\rm det}(\partial g))^n $$ works.

As per the "Additional remark" in my second answer to this MO question, I had a vague recollection of seeing a math-physics paper with such a formula, but maybe my memory is faulty. So I think it's better to ask the experts.

  • 2
    $\begingroup$ I remember this arxiv.org/pdf/hep-th/9209083v2.pdf but the difference is that instead of the Haar measure there is an Itsykson-Zuber exponent, too. Then however you can apply differential operator to M or N to get "correlation functions" and then set M=N=0. Is that helpful? $\endgroup$ Dec 1, 2016 at 0:31
  • 1
    $\begingroup$ :)- why ask on MO when I could have walked down the hallway and knocked on your door...Thanks this looks really interesting. $\endgroup$ Dec 1, 2016 at 0:46
  • 1
    $\begingroup$ :) the paper does not use differential operators but with them the statement hopefully can be obtained easier. I have not checked details though. $\endgroup$ Dec 1, 2016 at 0:50
  • 1
    $\begingroup$ and here arxiv.org/pdf/hep-th/0502041.pdf in introduction there is some discussion 10 years later $\endgroup$ Dec 1, 2016 at 0:51
  • $\begingroup$ Still worth adding a formal answer to this question, even if it's been sorted! $\endgroup$ Dec 1, 2016 at 5:46

1 Answer 1


To expand on my comments, this paper https://arxiv.org/pdf/hep-th/9209083v2.pdf by Shatashvili deals with ``correlation functions'' of Haar unitary matrices of the form $$ \int_{U(N)}^{} d\mu(U) e_{}^{tr(UAU_{}^{-1}B)} U_{i_1j_1}^{}\bar U_{k_1\ell_1}^{}\ldots U_{i_mj_m}^{}\bar U_{k_m\ell_m}, $$ and provides a certain combinatorial formula for these. Then setting $A=0$ would probably recover what you're asking about.

The same correlation functions (and an alternative formula for them) are also discussed in https://arxiv.org/pdf/hep-th/0502041.pdf


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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