From the $B$-Toda to the BKP hierarchy

TL;DR

This paper demonstrates that all BKP hierarchy tau-functions can be expressed as Pfaffians, classifies them via the universal orthogonal Grassmannian, and connects the KdV hierarchy as a special reduction of BKP, using mainly linear algebra techniques.

Contribution

It provides a new Pfaffian representation for BKP tau-functions, classifies these functions through the universal orthogonal Grassmannian, and links KdV as a 4-reduction of BKP, with explicit parameterizations.

Findings

BKP tau-functions are Pfaffians of skew-symmetric matrices.

Classification of tau-functions via the universal orthogonal Grassmannian.

KDV hierarchy is a 4-reduction of BKP hierarchy.

Abstract

It is shown that all $\tau$-functions of BKP hierarchy can be written as Pfaffians of skew-symmetric matrices. $\tau$-functions of BKP hierarchy are parameterized by points in the universal orthogonal Grassmannian manifold (UOGM). The UOGM is a disjoint union of Schubert cells, we classify and give explicit parameterization for points in each Schubert cell by constructing a frame for UOGM in the sense of Sato. $\tau$-functions are then expressed in terms of these frames and Schur-Q functions. For concreteness we give a comprehensive study for the $\tau$-functions of $B$-Toda which can be viewed as a finite version of the BKP hierarchy. Along the way we also give a constructive description for complex pure spinors du E. Cartan. As an application of our construction, we reprove a theorem due to A. Alexandrov which states that KdV solves BKP up to rescaling of the time parameters by $2$. We prove this by showing that the KdV hierarchy can be viewed as $4$-reduction of the BKP hierarchy. This interpretation gives complete characterization for the KdV orbits inside the BKP hierarchy. Other than a few facts from representation theory, the main tools we use to show the above results, however, are surprisingly simple linear algebra.