Wreath Macdonald operators

TL;DR

This paper introduces a new family of difference-permutation operators that are diagonalized by wreath Macdonald P-polynomials, linking algebraic structures with symmetric functions through integral formulas and quantum algebra representations.

Contribution

It constructs novel operators diagonalized by wreath Macdonald P-polynomials, connecting quantum toroidal algebra representations with symmetric functions.

Findings

Operators are diagonalized by wreath Macdonald P-polynomials.

Eigenvalues are expressed via elementary symmetric polynomials.

Operators originate from integral formulas in quantum algebra representations.

Abstract

We construct a novel family of difference-permutation operators and prove that they are diagonalized by the wreath Macdonald $P$-polynomials; the eigenvalues are written in terms of elementary symmetric polynomials of arbitrary degree. Our operators arise from integral formulas for the action of the horizontal Heisenberg subalgebra in the vertex representation of the corresponding quantum toroidal algebra