Complete set of cut-and-join operators in the Hurwitz–Kontsevich theory

We define cut-and-join operators in Hurwitz theory for merging two branch points of an arbitrary type. These operators have two alternative descriptions: (1) the $GL$ characters are their eigenfunctions and the symmetric group characters are their eigenvalues; (2) they can be represented as $W$-type differential operators (in particular, acting on the time variables in the Hurwitz–Kontsevich $\tau$-function). The operators have the simplest form when expressed in terms of the Miwa variables. They form an important commutative associative algebra, a universal Hurwitz algebra, generalizing all group algebra centers of particular symmetric groups used to describe the universal Hurwitz numbers of particular orders. This algebra expresses arbitrary Hurwitz numbers as values of a distinguished linear form on the linear space of Young diagrams evaluated on the product of all diagrams characterizing particular ramification points of the branched covering.