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Algebra of differential operators associated with Young diagrams. (English) Zbl 1242.22008
A correspondence between the Young diagrams and the algebra of differential operators of infinitely many variables is established. The differential operators W($$\Delta$$), corresponding to the Young diagrams $$\Delta$$, are closely related to the Hurwitz numbers, matrix integrals and integrable systems. It is proven that the Schur functions form a complete set of common eigenfunctions of these differential operators W($$\Delta$$), and their eigenvalues are expressed through the characters of symmetric groups. The algebra of Young diagrams, which is isomorphic to the algebra of conjugated classes of finite permutations of an infinite set is defined and its structure constants are expressed through the structure constants of the algebra $$A_n$$. A representation of the universal enveloping algebra U(gl($${\infty}))$$ in the algebra of differential operators of Miwa variables is constructed. Using this representation one associates with any Young diagram a differential operator of Miwa variables. This correspondence gives rise to an exact representation of the algebra $$A_n$$. An algorithm of calculating the operators W($$\Delta$$) is presented, and it is proven that the simplest non-trivial operator W($$[2]$$) is exactly the “cut-and-join” operator which plays an important role in the theory of Hurwitz numbers and moduli spaces. Then, the operators W($$\Delta$$) are interpreted as counterparts of the “cut-and-join” operator for the arbitrary Young diagram. It is proven that a special generating function of Hurwitz numbers satisfies a simple differential equation, which allows one to construct all the Hurwitz numbers successively.

##### MSC:
 22D20 Representations of group algebras 20B30 Symmetric groups 20E45 Conjugacy classes for groups 47L80 Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)
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