Unconstrained initial Hilbert-function sequences for Koszul Gorenstein algebras

Construct, for every sufficiently large integer d and every permutation π of {1, ..., ⌈d/2⌉}, an artinian Gorenstein Koszul algebra with Hilbert series ∑_{i=0}^d h_i t^i satisfying h_{π(1)} < ... < h_{π(⌈d/2⌉)}.

Background

The Roller Coaster theorem established in the paper shows that the first half of the Hilbert functions of certain artinian Gorenstein algebras can realize every ordering of their entries. The construction uses Nagata idealization of algebras associated with flag simplicial complexes, but it does not generally ensure that the resulting Gorenstein algebras are Koszul.

The conjecture asks whether the same unconstrained behavior persists within the narrower class of artinian Gorenstein Koszul algebras of large socle degree. The paper produces Koszul Gorenstein algebras failing the weak Lefschetz property, but leaves this stronger Roller Coaster statement unresolved.

References

Let $d \gg 0$ and $\pi$ be a permutation of the numbers ${1, \dots, \lceil d/2 \rceil}$. Then there exists an artinian Gorenstein Koszul algebra with Hilbert series $\sum_{i = 0}d h_i ti$ such that $$ h_{\pi(1)} < \dots < h_{\pi(\lceil d/2 \rceil)}. $$ In other words, the collection of sequences arising from the first half of the Hilbert series of artinian Gorenstein Koszul algebras of large socle degree is unconstrained.

— Roller Coaster Gorenstein algebras and Koszul algebras failing the weak Lefschetz property  (2502.00155 - Holleben et al., 31 Jan 2025) in Conjecture in Section 1 (Introduction), following Theorem 1.3; repeated as the final conjecture in Section 6