When are symmetric ideals monomial?

TL;DR

This paper investigates conditions under which symmetric ideals generated by polynomial orbits become monomial or have monomial radicals, providing specific cases and implications for infinite polynomial rings.

Contribution

It establishes new criteria for symmetric ideals to be monomial or have monomial radicals, including cases with homogeneous polynomials and square-free terms.

Findings

Ideals with generators containing variable powers have monomial radicals.

Polynomials with only square-free terms and non-zero sum of coefficients generate square-free monomial ideals.

Results extend to symmetric ideals in infinite polynomial rings.

Abstract

We study conditions on polynomials such that the ideal generated by their orbits under the symmetric group action becomes a monomial ideal or has a monomial radical. If the polynomials are homogeneous, we expect that such an ideal has a monomial radical if their coefficients are sufficiently general with respect to their supports. We prove this for instance in the case where some generator contains a power of a variable. Moreover, if the polynomials have only square-free terms and their coefficients do not sum to zero, then in a larger polynomial ring the ideal itself is square-free monomial. This has implications also for symmetric ideals of the infinite polynomial ring.