Symmetric ideals of the infinite polynomial ring

Let $R=mathbf{C}[xi_1,xi_2,ldots]$ be the infinite variable polynomial ring, equipped with the natural action of the infinite symmetric group $mathfrak{S}$. We classify the $mathfrak{S}$-primes of $R$, determine the containments among these ideals, and describe the equivariant spectrum of $R$. We emphasize that $mathfrak{S}$-prime ideals need not be radical, which is a primary source of difficulty. Our results yield a classification of $mathfrak{S}$-ideals of $R$ up to copotency. Our work is motivated by the interest and applications of $mathfrak{S}$-ideals seen in recent years.