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?

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?

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 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?

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?