Diagonal Solutions to the 2-Toda Hierarchy

TL;DR

This paper characterizes diagonal solutions to the 2-Toda hierarchy using combinatorial methods, showing that certain generating series and integrals are solutions to these complex differential equations.

Contribution

It introduces a combinatorial approach to describe solutions of the 2-Toda hierarchy and proves that content-type series are solutions, connecting various generating functions.

Findings

Content-type series are solutions to the 2-Toda hierarchy.

Permutation factorization generating series satisfy the hierarchy.

The HCIZ integral's character expansion is a solution.

Abstract

Using a combinatorial description of the Bernstein operator and its action on Schur functions, we describe the formal power series solutions to a family of partial differential equations known as the 2-Toda hierarchy. We also characterize diagonal solutions and use this to prove that a special family of formal power series, called content-type series, are solutions to the 2-Toda hierarchy. As examples, we prove that various generating series for permutation factorization problems (including the double Hurwitz problem), correlators for the Schur measure on partitions and the formal character expansion of the HCIZ integral are all solutions to the 2-Toda hierarchy.