ON THE NONPRIMALITY OF CERTAIN SYMMETRIC IDEALS

Abstract

Let $R=k[x_1,\dots ,x_n,\dots ]$ 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, Nagpal and Snowden [5] gave an indirect proof that the ${\mathfrak{𝔖}}_{\infty }$-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.