On symmetric partial differential operators

TL;DR

This paper introduces explicit second-order differential operators in the Weyl algebra that characterize trace functions related to symmetric polynomials, providing a holonomic system with solutions linked to roots of universal equations.

Contribution

It constructs a finite family of differential operators generating the ideal annihilating all trace functions, extending symmetric functions theory to differential operators.

Findings

Explicit differential operators for trace functions are provided.

A holonomic system related to roots of universal equations is constructed.

The approach generalizes symmetric functions theorem to differential operators.

Abstract

Let s 1 ,. .. , s k be the elementary symmetric functions of the complex variables x 1 ,. .. , x k. We say that F $\in$ C[s 1 ,. .. , s k ] is a trace function if their exists f $\in$ C[z] such that F (s 1 ,. .. , s k ] = k j=1 f (x j) for all s $\in$ C k. We give an explicit finite family of second order differential operators in the Weyl algebra W 2 := C[s 1 ,. .. , s k ] $\partial$ $\partial$s 1 ,. .. , $\partial$ $\partial$s k which generates the left ideal in W 2 of partial differential operators killing all trace functions. The proof uses a theorem for symmetric differential operators analogous to the usual symmetric functions theorem and the corresponding map for symbols. As a corollary, we obtain for each integer k a holonomic system which is a quotient of W 2 by an explicit left ideal whose local solutions are linear combinations of the branches of the multivalued root of the universal equation of degree k: z k + k h=1 (--1) h .s h .z k--h = 0.