Bihomogeneous symmetric functions

TL;DR

This paper introduces a differential operator on the space of symmetric functions with two gradings, providing a basis where the operator is triangular and enabling eigenvalue computation.

Contribution

The authors define a new operator T that preserves bihomogeneous components and construct a basis to facilitate eigenvalue analysis.

Findings

Eigenvalues of T are non-negative integers

A basis of bihomogeneous symmetric functions is constructed

Operator T is triangular in this basis

Abstract

We consider two natural gradings on the space of symmetric functions: by degree and by length. We introduce a differential operator $T$ that leaves the components of this double grading invariant and exhibit a basis of bihomogeneous symmetric functions in which this operator is triangular. This allows us to compute the eigenvalues of $T$, which turn out to be non-negative integers.