  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.
