Extended Schur's $Q$-functions and the full Kostant--Toda hierarchy on the Lie algebra of type $D$

TL;DR

This paper extends Schur's Q-functions to describe explicit polynomial tau-functions for the full Kostant--Toda hierarchy on the Lie algebra of type D, using Pfaffian invariants.

Contribution

It introduces an extension of Schur's Q-functions tailored for the type D Lie algebra and provides explicit formulas for the hierarchy's tau-functions.

Findings

Explicit Pfaffian-based flow formulas for the hierarchy.

New extended Schur's Q-functions for type D.

Explicit polynomial tau-function formulas.

Abstract

The full Kostant--Toda hierarchy on a semisimple Lie algebra is a system of Lax equations, in which the flows are determined by the gradients of the Chevalley invariants.This paper is concerned with the full Kostant--Toda hierarchy on the even orthogonal Lie algebra. By using a Pfaffian of the Lax matrix as one of the Chevalley invariants, we construct an explicit form of the flow associated to this invariant. As a main result, we introduce an extension of the Schur's $Q$-functions in the time variables, and use them to give explicit formulas for the polynomial $\tau$-functions of the hierarchy.