The Hirota Identity for Hyperpfaffian $\tau$-Functions in Charge-$L$ Ensembles

Abstract

We study log-gas ensembles with inverse temperature $\beta = L^2$ using a confluent Vandermonde representation that admits a formulation in the exterior algebra of a finite-dimensional vector space. By interpreting the system as consisting of finitely many particles with integer charge $L$, partition functions can be expressed exactly as hyperpfaffians. In this formulation, the system is governed by a natural momentum grading arising from the confluent Vandermonde structure, and its statistical observables are determined entirely by the corresponding bigraded commutative subalgebra. The geometric identity that a particle's $L$-blade wedges with itself to zero produces momentum Pl\"ucker relations within this algebra. These relations generate momentum transport identities between sectors of different particle number. Upon introducing dynamic time variables, the partition functions become $\tau$-functions, and these transport identities are transformed into Hirota bilinear equations. This provides an explicit algebraic origin for the integrable hierarchy structure of the $\beta = L^2$ ensembles, which may be viewed as a finite-dimensional analogue of the Sato Grassmannian formulation of integrable systems.