# integral kernel function for the SU(N) group

It is well know that the Haar probability measure for the $$U(N)$$ group, given by \begin{align} dX_{U(N)} & = \frac{1}{N!(2\pi)^N} \begin{vmatrix} 1 & 1 & \cdots & 1 & 1 \\ e^{i\lambda_1} & e^{i\lambda_2} & \cdots & e^{i\lambda_{N-1}} & e^{i\lambda_N} \\ e^{i2\lambda_1} & e^{i2\lambda_2} & \cdots & e^{i2\lambda_{N-1}} & e^{i2\lambda_N} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ e^{i(N-1)\lambda_1} & e^{i(N-1)\lambda_2} & \cdots & e^{i(N-1)\lambda_{N-1}} & e^{i(N-1)\lambda_N} \\ \end{vmatrix}^2 d\lambda_1\dots d\lambda_N \\ & = \frac{1}{N!(2\pi)^N} \prod_{1\leq j can also be expressed as a determinantal point process $$dX_{U(N)} = \frac{1}{N!(2\pi)^N} \det_{NXN}(S_N(\lambda_j,\lambda_k))_{1\leq j,k\leq N} \text{ } d\lambda_1\dots d\lambda_N$$ where $$S_N(x,y)=\frac{\sin\frac{N(x-y)}{2}}{\sin\frac{x-y}{2}}$$ is the integral kernel function for the $$U(N)$$ group (a proof of this fact can be found, for example, in section 4.1 of this paper); similar kernels have also been found for the special orthogonal and symplectic groups.

My question is whether any analogous kernel function is know for the $$SU(N)$$ group ? (or if it can be proved that it doesn't exist).

If such a function $$K(x,y)$$ exists, it would allow one to express the $$SU(N)$$ Haar probability measure \begin{align} dX_{SU(N)} & = \frac{1}{N!(2\pi)^{N-1}} \begin{vmatrix} 1 & \cdots & 1 & 1 \\ e^{i\lambda_1} & \cdots & e^{i\lambda_{N-1}} & e^{-i(\lambda_1+\ldots+\lambda_{N-1})} \\ e^{i2\lambda_1} & \cdots & e^{i2\lambda_{N-1}} & e^{-i2(\lambda_1+\ldots+\lambda_{N-1})} \\ \vdots & \ddots & \vdots & \vdots \\ e^{i(N-1)\lambda_1} & \cdots & e^{i(N-1)\lambda_{N-1}} & e^{-i(N-1)(\lambda_1+\ldots+\lambda_{N-1})} \\ \end{vmatrix}^2 d\lambda_1\dots d\lambda_{N-1} \\ & = \frac{1}{N!(2\pi)^{N-1}} \prod_{1\leq j in the form of $$dX_{SU(N)} = \frac{1}{N!(2\pi)^{N-1}} \det_{(N-1)X(N-1)}(K(\lambda_j,\lambda_k))_{1\leq j,k\leq N-1} \text{ } d\lambda_1\dots d\lambda_{N-1}$$

## 1 Answer

The integral kernel for $${\rm U}\,(N)$$, due to Dyson, has been generalized by Katz and Sarnak to other compact groups (Random Matrices, Frobenius Eigenvalues, and Monodromy, page 121). Their result has the general form $$d\mu=\frac{1}{n!}\det_{n\times n}[L_N(\lambda_i,\lambda_j)]\prod_{i=1}^{n}\frac{d\lambda_i}{\sigma\pi},\;\;\lambda_i\in[0,\sigma\pi],\;\;1\leq i\leq n,$$ $$S_N(x)=\frac{\sin(Nx/2)}{\sin(x/2)},\;\; L_N(x,y)=\tfrac{1}{2}\sigma[S_{\rho N+\tau}(x-y)+\varepsilon S_{\rho N+\tau}(x+y)].$$ The coefficients are tabulated as follows:

Note that $$\mu[{\rm O}_-(2N+1)]$$ is the same as $$\mu[{\rm SO}(2N+1)]$$, since the matrices differ by a minus sign. Also note that $$\mu[{\rm O}_-(2N+2)]=\mu({\rm U\,Sp}(2N)]$$.

The group $${\rm SU}\,(N)$$ is conspicuously missing from this table... I would assume there is a reason for this (Katz and Sarnak discuss $${\rm SU}\,(N)$$ at various other points in their text). My surmise is that there is no way to incorporate the delta function $$\delta\bigl(\sum_{i=1}^N \lambda_i\bigr)$$ into an $$(N-1)\times(N-1)$$ determinant.

• Thank you for your answer ! I also suspect the same thing, that it's going to turn out that the SU(N) kernel doesn't exist, but hopefully it should be possible to prove this in a rigorous way. – Amadocta Sep 24 '18 at 20:31