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Young diagrams can be parameterized with the help of hook variables, which iswell known but never studied in big detail. We demonstrate that this is themost adequate parameterization for many physical applications: from the Schurfunctions, conventional, skew and shifted, which all satisfy their own kinds ofdeterminant formulas in these coordinates, to KP/Toda integrability and relatedbasis of cut-and-join $\hat W$-operators, which are both actually expressedthrough the single-hook diagrams. In particular, we discuss a new type ofmulti-component KP $\tau$-functions, Matisse $\tau$-functions. We alsodemonstrate that the Casimir operators, which are responsible forintegrability, are single-hook, with the popular basis of "completed cycles"being distinguished by especially simple coefficients in the correspondingexpansion. The Casimir operators also generate the $q=t$ RuijsenaarsHamiltonians. However, these properties are broken by the naive Macdonalddeformation, which is the reason for the loss of KP/Toda integrability andrelated structures in $q$-$t$ matrix models.

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