Hook variables: cut-and-join operators and $\tau$-functions

TL;DR

This paper explores the use of hook variables for parameterizing Young diagrams, revealing their significance in KP/Toda integrability, cut-and-join operators, and related physical applications, while also discussing limitations under Macdonald deformation.

Contribution

It demonstrates that hook variables are the most suitable parameterization for various functions and operators in integrable systems, providing new insights into their structure and properties.

Findings

Hook variables effectively parameterize Schur functions and related determinants.

Casimir operators are single-hook and generate Ruijsenaars Hamiltonians.

Macdonald deformation breaks KP/Toda integrability and related structures.

Abstract

Young diagrams can be parameterized with the help of hook variables, which is well known but never studied in big detail. We demonstrate that this is the most adequate parameterization for many physical applications: from the Schur functions, conventional, skew and shifted, which all satisfy their own kinds of determinant formulas in these coordinates, to KP/Toda integrability and related basis of cut-and-join $\hat W$-operators, which are both actually expressed through the single-hook diagrams. In particular, we discuss a new type of multi-component KP $\tau$-functions, Matisse $\tau$-functions. We also demonstrate that the Casimir operators, which are responsible for integrability, are single-hook, with the popular basis of "completed cycles" being distinguished by especially simple coefficients in the corresponding expansion. The Casimir operators also generate the $q=t$ Ruijsenaars Hamiltonians. However, these properties are broken by the naive Macdonald deformation, which is the reason for the loss of KP/Toda integrability and related structures in $q$-$t$ matrix models.