General symmetric ideals

TL;DR

This paper studies the structure and properties of symmetric ideals generated by general forms, introducing a new parametrization method and demonstrating their extremal behavior and stability in algebraic invariants.

Contribution

It introduces a bijective parametrization of symmetric ideals using Macaulay-Matlis duality and shows their extremal and stable properties as the number of variables grows.

Findings

Symmetric ideals exhibit extremal Hilbert functions and Betti numbers.

They satisfy the Weak Lefschetz Property.

Algebraic and homological invariants stabilize asymptotically.

Abstract

We investigate the structure and properties of symmetric ideals generated by general forms in the polynomial ring under the natural action of the symmetric group. This work significantly broadens the framework established in our earlier collaboration with Harada on principal symmetric ideals. A novel aspect of our approach is the construction of a bijective parametrization of general symmetric ideals using Macaulay-Matlis duality, which is asymptotically independent of the number of variables of the ambient ring. We establish that general symmetric ideals exhibit extremal behavior in terms of Hilbert functions and Betti numbers, and satisfy the Weak Lefschetz Property. We also demonstrate explicit asymptotic stability in their algebraic and homological invariants under increasing numbers of variables, showing that such ideals form well-behaved $\mathfrak{S}_\infty$-invariant chains.