On ideals generated by two generic quadratic forms in the exterior algebra

TL;DR

This paper investigates the algebraic structure of ideals generated by two generic quadratic forms in the exterior algebra, proposing a conjecture for their Hilbert series and providing partial proofs and bounds.

Contribution

It introduces a conjecture for the Hilbert series of such ideals and establishes it as an upper bound, also relating it to squarefree polynomial rings.

Findings

Conjectured Hilbert series as an upper bound

Determined majority of the coefficients of the series

Proposed equivalence with series of squarefree polynomial rings

Abstract

Based on the structure theory of pairs of skew-symmetric matrices, we give a conjecture for the Hilbert series of the exterior algebra modulo the ideal generated by two generic quadratic forms. We show that the conjectured series is an upper bound in the coefficient-wise sense, and we determine a majority of the coefficients. We also conjecture that the series is equal to the series of the squarefree polynomial ring modulo the ideal generated by the squares of two generic linear forms.