On the non primality of certain symmetric ideals

TL;DR

This paper provides a direct proof that certain symmetric ideals in an infinite variable polynomial ring are not prime, advancing understanding of their algebraic structure and proposing potential generalizations.

Contribution

It offers the first explicit element-based proof of non-primality for specific symmetric ideals and suggests conjectures for broader cases.

Findings

Explicit elements demonstrating non-primality

Direct proof contrasting previous indirect methods

Formulation of conjectures for generalizations

Abstract

Let $R =k[x_1, \cdots, x_n,\cdots]$ be the infinite variable polynomial ring equipped with the natural $\mathfrak{S}_\infty$ action, where $k$ is a field of characteristic zero. In recent work \cite{NS21}, Nagpal--Snowden gave an indirect proof that $\mathfrak{S}_\infty$-ideal generated by $(x_1-x_2)^{2n}$ is not $\mathfrak{S}_\infty$-prime. In this paper, we give a direct proof, with explicit elements. We further formulate some conjectures on possible generalizations of the result.