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On the non primality of certain symmetric ideals (2112.10207v1)

Published 19 Dec 2021 in math.AC

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.

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