Let $\lambda_1, \dots, \lambda_n$ be the eigenvalues of a random Unitary matrix. I am interested in the expected value:

$$\mathbb{E}_{X \in U(N)}\left[ \prod_{i=1}^n \frac{1}{(1 - \lambda_i)^2}\right]$$

Symmetric functions of eigenvalues can be thought of as characters as some representation, $\rho$ and we are counting how many copies of the identity representation in there:

$$ \rho = 1 \oplus \dots $$

It looks like my product is the resolvent of the ~~regular~~ *defining* representation.

$$ \det_R \ (1-X)^{-2} $$

Can we evaluate this matrix integral directly using properties of the regular representation or is it easier to use the Weyl integration formula?

resolventof the regular representation. And then you take the determinant of that - which is the product of the eigenvalues. If we set $x = e^{2\pi i t}$ we get the Weyl integration formula, where $$ f(x) = \prod_{i=1}^n \frac{1}{(1 - x_i)^2}$$ So then I asked myself "Which representation is this?" $\endgroup$9more comments