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

We give a hyperpfaffian formulation for correlation functions in $β$-ensembles of M × M random matrices when β= L² is an even square integer. More specifically, to the mth correlation function Rₘ : ᵐ → [0, ∞) we associate the L-vector valued function ωₘ : ᵐ → ΛL L(M-m) such that Rₘ( y) is given by the Vandermonde determinant in y₁, …, yM times the hyperpfaffian of ωₘ. The partition function of the ensemble was previously shown to be the hyperpfaffian of a { Gram} L-form $ω$ in ΛL LM, and we demonstrate the relationship between ωₘ( y) and $ω$, 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 ω( y) so it is very sparse (relative to the expected L(M-m) L coefficients of a general L-vector). These generalize skew-orthogonal polynomials arising in the well-understood $β= 4$ situation. Finally we explore the situation in the circular β= L² ensembles. Here the monomials give a prototype, and we give explicit formulas for (the circular versions of) $ω$ and ωₘ. We use our hyperpfaffian framework to produce exact formulas for the two point function when $β= 16$ for small values $M.$ Along the way we will record hyperpfaffian evaluations using known values of partition functions of $β$-ensembles.