Coinvariant algebraic g-conjecture

Determine whether the coinvariant artinian algebra associated with every K-homology sphere satisfies the strong Lefschetz property; equivalently, establish whether, for a general linear form ell, the dimension of each ell-coinvariant stress space equals the positive part of the difference between consecutive graded dimensions of the coinvariant algebra.

Background

The paper introduces the coinvariant algebra of a simplicial complex Delta as the quotient by its Stanley–Reisner ideal together with the first d+1 elementary symmetric polynomials, where Delta is d-dimensional. For a K-homology sphere, this quotient is an Artinian Gorenstein algebra whose top coinvariant stress has an explicit Vandermonde-type formula.

The strong Lefschetz property for suitable linear artinian reductions is closely related to the classical g-conjecture for simplicial spheres. The authors ask whether the analogous Lefschetz statement holds for the universal, nonlinear coinvariant reduction, which would extend the known linear-setting result and connect coinvariant stresses with Lefschetz theory.

References

Let $\Delta$ be a $K$-homology sphere. Does the ring $(\Delta)$ satisfy the SLP? In other words, for a general linear form $\ell$, does the following equality hold? $$ \dim S_k(\Delta, \ell) = \max(\dim (\Delta)k - \dim (\Delta){k - 1}, 0) $$

— Coinvariant stresses, Lefschetz properties and random complexes  (2501.12108 - Holleben, 21 Jan 2025) in Question 4.1, Section 8, “Concluding remarks and future work”