B\"acklund transformations for certain rational solutions of Painlev\'e VI

TL;DR

This paper introduces Bäcklund transformations for rational solutions of Painlevé VI, derived from KP Grassmannian reductions, connecting root lattices of sl(6) and F4^{(1)} to relate tau functions.

Contribution

It presents new Bäcklund transformations for Painlevé VI rational solutions based on Grassmannian reductions, linking root lattices of sl(6) and F4^{(1)}.

Findings

Transformations act on Painlevé VI tau functions.

Root lattice of sl(6) related to F4^{(1)}.

Transformations connect different tau function parametrizations.

Abstract

We introduce certain B\"acklund transformations for rational solutions of the Painlev\'e VI equation. These transformations act ona family of Painlev\'e VI tau functions. They are obtained from reducing the Hirota bilinear equations that describe the relation between certain points in the 3 component polynomial KP Grassmannian. In this way we obtain transformations that act on the root lattice of sl(6). We also show that this sl(6) root lattice can be related to the $F_4^{(1)}$ root lattice. We thus obtain B\"acklund transformations that relate Painlev\'e VI tau functions, parametrized by the elements of this $F_4^{(1)}$ root lattice.