On generic principal ideals in the exterior algebra

TL;DR

This paper establishes a lower bound on the Hilbert series of exterior algebra quotients by generic odd-degree forms, disproves a related conjecture, and identifies cases where the bound is minimal.

Contribution

It provides a new lower bound for the Hilbert series in exterior algebra modulo principal ideals generated by generic forms and refutes a previous conjecture.

Findings

Lower bound on Hilbert series established

Conjecture by Moreno-Socías and Snellman disproved

Minimal Hilbert series identified in specific cases

Abstract

We give a lower bound on the Hilbert series of the exterior algebra modulo a principal ideal generated by a generic form of odd degree and disprove a conjecture by Moreno-Soc\'ias and Snellman. We also show that the lower bound is equal to the minimal Hilbert series in some specific cases.