Multiple sums and integrals as neutral BKP tau functions

TL;DR

This paper demonstrates how multiple sums and integrals can be represented as tau functions of the BKP hierarchy using neutral fermions, linking combinatorial sums over strict partitions to integrable systems.

Contribution

It introduces a novel representation of sums and integrals as BKP tau functions, connecting projective Schur functions to integrable hierarchies.

Findings

Discrete analogues of beta-ensembles for β=1,2,4

Representation of sums as BKP tau functions

Connection between sums over strict partitions and integrable systems

Abstract

We consider multiple sums and multi-integrals as tau functions of the BKP hierarchy using neutral fermions as the simplest tool for deriving these. The sums are over projective Schur functions $Q_\alpha$ for strict partitions $\alpha$. We consider two types of such sums: weighted sums of $Q_\alpha$ over strict partitions $\alpha$ and sums over products $Q_\alpha Q_\gamma$. In this way we obtain discrete analogues of the beta-ensembles ($\beta=1,2,4$). Continuous versions are represented as multiple integrals. Such sums and integrals are of interest in a number of problems in mathematics and physics.