Pfaffian and determinantal tau functions I

TL;DR

This paper provides a representation theoretical explanation for the relationship between Pfaffian and determinantal tau functions, showing how squares of certain tau functions relate to two-component KP tau functions.

Contribution

It introduces a new representation theoretical framework connecting Pfaffian and determinantal tau functions through 2-BKP and 2-KP hierarchies.

Findings

Square of a BKP tau function equals a two-component KP tau function

Square of a 2-BKP tau function equals a two-component 2-KP tau function

Provides a theoretical explanation for the Pfaffian-Toda tau function relationship

Abstract

Adler, Shiota and van Moerbeke observed that a tau function of the Pfaff lattice is a square root of a tau function of the Toda lattice hierarchy of Ueno and Takasaki. In this paper we give a representation theoretical explanation for this phenomenon. We consider 2-BKP and two-component 2-KP tau functions. We shall show that a square of a BKP tau function is equal to a certain two-component KP tau function and a square of a 2-BKP tau function is equal to a certain two-component 2-KP tau function.