Symmetric and Isotypic Hilbert Series for Symmetric Ideals

An ideal in a polynomial ring is symmetric if it is invariant under any permutation of variables. In this paper, we define and study the symmetric and isotypic Hilbert series for symmetric ideals in a polynomial ring with countably many variables. The symmetric Hilbert series is the limit of the Hilbert series of the invariant parts of the finite truncated quotients, while the isotypic Hilbert series records stable multiplicities of irreducible symmetric-group representations for each degree. Our main result proves that, under a mild support condition on the ideal, the symmetric Hilbert series is a rational function. We further show that this rationality extends to the isotypic Hilbert series for every irreducible representation. The proofs of these results rely on the monomial structure of the polynomials within the symmetric ideal, combined with Kostka inversion for the isotypic case.