On the Solvability of Beta-Ensembles when Beta is a Square Integer

Abstract

We use combinatorial identities in the shuffle and exterior algebra to present hyperpfaffian formulations of partition functions for ß-ensembles with arbitrary probability measure when ß is a square integer. This is an analogue of the de Bruijn integral identities for the ß = 1 and ß = 4 ensembles. We also generalize several classic algebraic identities for determinants and Pfaffians to identities for Hyperpfaffians, extending the fermionic and bosonic Wick formulas which frequently arise in Quantum Field Theory.