Unconstrained first-half Hilbert functions for Koszul Gorenstein algebras

Construct, for every sufficiently large socle degree d and every permutation $\pi$ of $\{1,\ldots,\lceil d/2\rceil\}$, an artinian Gorenstein Koszul algebra with Hilbert series $\sum_{i=0}^{d}h_i t^i$ satisfying $h_{\pi(1)}<\cdots<h_{\pi(\lceil d/2\rceil)}$, thereby proving that the first-half Hilbert-function sequences of artinian Gorenstein Koszul algebras of large socle degree are unconstrained.

Background

The paper proves an analogous Roller Coaster theorem for artinian Gorenstein algebras and later establishes unconstrained first-half Hilbert functions for flag simplicial Perazzo algebras. It also constructs Koszul Gorenstein algebras failing the weak Lefschetz property from whiskered bipartite graphs.

The remaining conjecture asks whether the unconstrained Hilbert-function phenomenon can be achieved while retaining the Koszul property for every sufficiently large socle degree and every prescribed ordering of the first half of the Hilbert function.

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 the discussion of Migliore–Nagel’s conjecture