Hook variables: cut-and-join operators and τ-functions

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.