BKP tau-functions as square roots of KP tau-functions

TL;DR

This paper explores the relationship between BKP and KP tau-functions, providing explicit computations and a representation-theoretical proof of their connection, especially in polynomial cases.

Contribution

It explicitly computes the KP square root of polynomial BKP tau-functions and proves a recent result linking KdV and BKP tau-functions through division of times.

Findings

Polynomial BKP tau-functions have explicit KP square roots.

A representation-theoretical proof relates KdV and BKP tau-functions.

Dividing KdV times by 2 transforms KdV tau-functions into BKP tau-functions.

Abstract

It is well-known that a BKP tau-function is the square root of a certain KP tau-function, provided one puts the even KP times equal to zero. In this paper we compute for all polynomial BKP tau-function its corresponding KP "square". We also give, in the polynomial case, a representation theoretical proof of a recent result by Alexandov, viz. that a KdV tau-function becomes a BKP tau-function when one divides all KdV times by 2.