The Lefschetz and Hodge–Riemann strengthening of Almkvist's conjecture

Let $m,n$ be positive integers and define the graded Artinian Gorenstein algebra

$$A(m,n)=\frac{{\mathbb Q}[e_1,\ldots,e_n]}{(e_1(m),\ldots,e_n(m))}.$$

If $A(m,n)$ has socle degree $d$, its strong Lefschetz property (SLP) means that there is a linear form $\ell\in A_1$ for which

$$\times\ell^{d-2i}\colon A_i\longrightarrow A_{d-i},\qquad 0\leq i\leq\left\lfloor\frac d2\right\rfloor,$$

are all isomorphisms. The Lefschetz property with respect to the Hilbert function (HLP) means that these maps have rank equal to the minimum dimension of the graded components between $i$ and $d-i$. The complex Hodge–Riemann relations (complex HRR) mean that these maps have signature equal to the alternating sum of the even first differences of the Hilbert function.

Almkvist–Lefschetz strengthening. (i) For each $m$, the algebra $A(m,n)$ satisfies the SLP for all $n\geq 11$. (ii) Moreover, if $m$ is even, then $A(m,n)$ satisfies the SLP for all $n$. (iii) For each $m$, the algebra $A(m,n)$ satisfies HLP for all $n$. (iv) If $m$ is even, then $A(m,n)$ satisfies the complex HRR.

SLP implies unimodality of the Hilbert function, so the first two assertions strengthen Almkvist's conjecture; the latter assertions concern finer Lefschetz and signature properties. The source states that items (i)–(iii) were posed in earlier work and item (iv) is new, but gives no resolution of these assertions.