Hyperpfaffian Correlations for Beta-Ensembles: Beta an Even Square Integer

We give a hyperpfaffian formulation for correlation functions in $\beta$-ensembles of $M\times M$ random matrices when $\beta =L^2$ is an even square integer. More specifically, to the $m$th correlation function $R_m:{\mathbb{R}}^m\to [0,\mathrm{\infty })$ we associate the $L$-vector valued function ${\omega }_m:{\mathbb{R}}^m\to {\mathrm{\Lambda }}^L{\mathbb{R}}^{L(M-m)}$ such that $R_m(\mathbf{y})$ is given by the Vandermonde determinant in $y_1,\dots ,y_M$ times the hyperpfaffian of ${\omega }_m\mathrm{.}$ The partition function of the ensemble was previously shown to be the hyperpfaffian of a {\it Gram} $L$-form $\omega$ in ${\mathrm{\Lambda }}^L{\mathbb{R}}^{LM},$ and we demonstrate the relationship between ${\omega }_m(\mathbf{y})$ and $\omega$, both having coefficients built from integrals of Wronskians of monic polynomials. Assuming the existence of families of polynomials sympathetic with the weight of the ensemble, we may construct $\omega (\mathbf{y})$ so it is very sparse (relative to the expected $\left(\genfrac{}{}{0px}{}{L(M-m)}{L}\right)$ coefficients of a general $L$-vector). These generalize skew-orthogonal polynomials arising in the well-understood $\beta =4$ situation. Finally we explore the situation in the circular $\beta =L^2$ ensembles. Here the monomials give a prototype, and we give explicit formulas for (the circular versions of) $\omega$ and ${\omega }_m\mathrm{.}$ We use our hyperpfaffian framework to produce exact formulas for the two point function when $\beta =16$ for small values $M\mathrm{.}$ Along the way we will record hyperpfaffian evaluations using known values of partition functions of $\beta$-ensembles.