Generic Jordan type of homology-sphere reductions

Characterize the ranks and full-rank failures of all multiplication maps by powers of the sum of the variables in the monomial artinian reduction associated with a rational homology sphere; specifically, determine whether exactly one multiplication map by the sum of the variables fails to have full rank and whether its rank defect is exactly one.

Background

The paper studies the algebra J = I_Delta + (x_1{d+2},...,x_n{d+2}) for a d-dimensional rational homology sphere Delta, and proves that this algebra fails the weak Lefschetz property. The failure arises from a nonzero top coinvariant stress in the kernel of a multiplication map by L=x_1+...+x_n.

Computational evidence reported by the authors suggests that the failure may be highly constrained rather than occurring in many degrees or with large rank defects. The proposed problem seeks a complete description of the Jordan type, or equivalently the ranks of multiplication by all powers of L, for these monomial reductions.

References

Let $\Delta$ be a $d$-dimensional $Q$-homology sphere and $J = I_\Delta + (x_1{d+2}, \dots, x_n{d+2})$. Then is there a unique multiplication map by $L = x_1 + \dots + x_n$ that fails to have full rank? Does this map only fail by $1$? More generally, what can be said about the ranks of the maps $\times Li : \frac{R}{J} \to \frac{R}{J}$?

— Coinvariant stresses, Lefschetz properties and random complexes  (2501.12108 - Holleben, 21 Jan 2025) in Question 4.3, Section 8, “Concluding remarks and future work”