Pfaffian and Determinant Solutions to A Discretized Toda Equation for $B_r, C_r$ and $D_r$

TL;DR

This paper presents Pfaffian and determinant solutions to a discretized Toda equation related to classical Lie algebras, connecting integrable systems with algebraic structures like Yangians and Schur functions.

Contribution

It introduces explicit Pfaffian and determinant solutions for Toda equations associated with $B_r, C_r, D_r$ Lie algebras, extending classical formulas to a discrete setting.

Findings

Solutions expressed via Pfaffians and determinants

Connections to Yangian analogues of Jacobi-Trudi formulas

Applicable to Toda equations for classical Lie algebras

Abstract

We consider a class of 2 dimensional Toda equations on discrete space-time. It has arisen as functional relations in commuting family of transfer matrices in solvable lattice models associated with any classical simple Lie algebra $X_r$. For $X_r = B_r, C_r$ and $D_r$, we present the solution in terms of Pfaffians and determinants. They may be viewed as Yangian analogues of the classical Jacobi-Trudi formula on Schur functions.