Discrete integrable systems and condensation algorithms for Pfaffians

TL;DR

This paper explores how discrete integrable systems can be used to develop condensation algorithms for Pfaffians, providing explicit methods with adjustable parameters and demonstrating their effectiveness.

Contribution

It introduces novel condensation algorithms for Pfaffians derived from discrete integrable systems, including explicit one-parameter methods and their connection to Pfaffian τ-functions.

Findings

Effective Pfaffian condensation algorithms developed

Explicit one-parameter algorithms constructed

Demonstrated connection to discrete integrable systems

Abstract

Inspired by the connection between the Dodgson's condensation algorithm and Hirota's difference equation, we consider condensation algorithms for Pfaffians from the perspectives of discrete integrable systems. The discretisation of Pfaffian elements demonstrate its effectiveness to the Pfaffian $\tau$-functions and discrete integrable systems. The free parameter in the discretisation allows us in particular to obtain explicit, one-parameter condensation algorithms for the Pfaffians.