On generic principal ideals in the exterior algebra

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-Socías and Snellman. We also show that the lower bound is equal to the minimal Hilbert series in some specific cases.

Algebraic Stories from One and from the Other Pockets

“…A lower bound for the series of E/( f ) is given in Lundqvist and Nicklasson (2018). It is also shown that the lower bound agrees with the generic series is some special cases.…”

“…Thus, the series cannot be equal to [(1 + t) n (1 − t d )] + when f has odd degree equal to d. The annihilator ideal Ann( f ) shows some unexpected behavior. Indeed, when (n, d) = (9, 3), the map induced by multiplication be a generic cubic form from E 3 (of dimension 9 3 ) to E 6 (of dimension 9 6 = 9 3 ) has a kernel of dimension four, see Lundqvist and Nicklasson (2018), while a one-dimensional kernel is what one would expect.…”

Questions and conjectures on extremal Hilbert series

“…In [22], the series for Λ(V )/(f ) is determined for d odd and equal to n − 2 (the expected series + t n−1 ) and n − 3 (the expected series) and for (9, 3) (the expected series + 3t 6 ). A lower bound for the series for any d is also given, providing a counterexample to Conjecture 6.1 in [28].…”