Cubature Rules for Unitary Jacobi Ensembles

Abstract

We present Chebyshev type cubature rules for the exact integration of rational symmetric functions with poles on prescribed coordinate hyperplanes. Here the integration is with respect to the densities of unitary Jacobi ensembles stemming from the Haar measures of the orthogonal and the compact symplectic Lie groups.

Appendix A. The Cauchy–Binet–Andréief Formulas

Let $f_1,\ldots f_n$ and $g_1,\ldots ,g_n$ be functions in the Hilbert space $L^2\bigl ( ]a,b[ , | \text {w}(x) | \text {d}x\bigr )$. Then Andréief’s integration formula states that [1]:

$$\begin{aligned}&\frac{1}{n!} \int _a^{b}\cdots \int _a^b \det \left[ f_j(x_k) \right] _{1\le j,k\le n} \det \left[ g_j(x_k)\right] _{1\le j,k\le n} \prod _{1\le j\le n} \text {w}(x_j)\, \text {d} x_1\cdots \text {d} x_n \nonumber \\&\qquad =\det \left[ \int _a^b f_j(x)g_k(x) \text {w}(x) \text {d}x \right] _{1\le j,k\le n} . \end{aligned}$$

(A.1)

A short and elementary verification of this identity is provided, e.g., in [2, Lemma 3.1]. A similar proof can be found in [13] together with a historical account of the formula. If we pass from functions on the interval to functions supported on the nodes $x_0^{(m+n)}<\cdots <x_{m+n-1}^{(m+n)}$, and replace the integration measure $\text {w}(x)\text {d}x$ by the discrete point measure on these nodes with weights $\text {w}_0^{(m+n)},\ldots ,\text {w}_{m+n-1}^{(m+n)}$, then Andréief’s integration formula gives rise to the identity

$$\begin{aligned}&\frac{1}{n!}\sum _{\begin{array}{c} 0\le l_k<m+n \\ k=1,\ldots ,n \end{array}} \Biggl ( \det \left[ f_j\bigl (x^{(m+n)}_{l_k} \bigr ) \right] _{1\le j,k\le n} \det \left[ g_j \bigl (x^{(m+n)}_{l_k} \bigr ) \right] _{1\le j,k\le n} \prod _{1\le j\le n} \text {w}^{(m+n)}_{l_j} \Biggr ) \\&\qquad =\det \left[ \sum _{0\le l< m+n} f_j \bigl (x^{(m+n)}_l\bigr ) g_k \bigl (x^{(m+n)}_l\bigr ) \text {w}^{(m+n)}_{l} \right] _{1\le j,k\le n} . \end{aligned}$$

Upon exploiting the anti-symmetry of the determinants with respect to the ordering of the indices $l_1,\ldots ,l_n$, the left hand side can be rewritten as

$$\begin{aligned} \sum _{m+n>l_1>l_2>\cdots >l_n\ge 0} \Biggl ( \det \left[ f_j\bigl (x^{(m+n)}_{l_k} \bigr ) \right] _{1\le j,k\le n} \det \left[ g_j \bigl (x^{(m+n)}_{l_k} \bigr ) \right] _{1\le j,k\le n} \prod _{1\le j\le n} \text {w}^{(m+n)}_{l_j} \Biggr ). \end{aligned}$$

This shows that the identity of interest amounts to the celebrated Cauchy–Binet formula

$$\begin{aligned} \sum _{\begin{array}{c} L\subset \{ 0,\ldots ,m+n-1\} \\ |L| = n \end{array}} \det F_L \det G_L = \det \left( F G^T \right) , \end{aligned}$$

(A.2)

with $F=F_{\{ 0,\ldots ,m+n-1\}}$, $G=G_{\{ 0,\ldots ,m+n-1\}}$ and

$$\begin{aligned} F_L= \left[ f_j\bigl (x^{(m+n)}_{l} \bigr ) \right] _{\begin{array}{c} 1\le j \le n\\ l\in L \end{array} } , \qquad G_L= \left[ g_j\bigl (x^{(m+n)}_{l} \bigr ) \text {w}^{(m+n)}_{l} \right] _{\begin{array}{c} 1\le j \le n\\ l\in L \end{array} } . \end{aligned}$$

In Eq. (A.2) the sum is over all subsets of indices L of cardinality $|L|=n$, and $G^T$ refers to the transposed matrix.