Algebra of differential operators associated with Young diagrams

TL;DR

This paper constructs a commutative algebra of differential operators linked to Young diagrams, revealing deep connections with symmetric group characters, Hurwitz numbers, and Schur functions as eigenfunctions.

Contribution

It introduces a novel algebraic framework connecting Young diagrams, differential operators, and symmetric group representations.

Findings

Schur functions are eigenfunctions of the constructed differential operators.

Eigenvalues are expressed via symmetric group characters.

Structure constants relate to Hurwitz numbers.

Abstract

We establish a correspondence between Young diagrams and differential operators of infinitely many variables. These operators form a commutative associative algebra isomorphic to the algebra of the conjugated classes of finite permutations of the set of natural numbers. The Schur functions form a complete system of common eigenfunctions of these differential operators, and their eigenvalues are expressed through the characters of symmetric groups. The structure constants of the algebra are expressed through the Hurwitz numbers.