The minimal free resolution of a generic symmetric principal ideal

TL;DR

This paper studies principal symmetric ideals generated by symmetric group orbits, showing their minimal free resolutions are generically constant and explicitly determined, with implications for certain classes of compressed artinian algebras.

Contribution

It introduces principal symmetric ideals, characterizes their generic minimal free resolutions, and explores related narrow algebra classes.

Findings

Minimal free resolution is constant on a Zariski open subset.

Explicit resolution determination for principal symmetric ideals.

Analysis of narrow and extremely narrow graded algebras.

Abstract

We introduce the class of principal symmetric ideals, which are ideals generated by the orbit of a single polynomial under the action of the symmetric group. Fixing the degree of the generating polynomial, this class of ideals is parametrized by points in a suitable projective space. We show that the minimal free resolution of a principal symmetric ideal is constant on a nonempty Zariski open subset of this projective space and we determine this resolution explicitly. Along the way, we study two classes of graded algebras which we term narrow and extremely narrow; both of which are instances of compressed artinian algebras.