weak Lefschetz property.

\[cor:one-map\] Assume the parameters of $I = I_{a,b,c,\alpha,\beta,\gamma}$ satisfy the conditions in Proposition \[pro:semistable\] and suppose $a + b + c + \alpha + \beta + \gamma \equiv 0 \pmod{3}$. Then $R/I$ has the weak Lefschetz property if and only if the map $\times(x+y+z): [R/I]_s \rightarrow [R/I]_{s+1}$ is injective (or surjective).

This follows immediately from Lemma \[lem:twin-peaks\] by using Propositions \[pro:surj\]–\[pro:mono\].

This leads to the definition of two matrices with determinants that determine the weak Lefschetz property. The first is a zero-one matrix and the second is a matrix of binomial coefficients.

\[pro:wlp-zero-one\] Assume the parameters of $I = I_{a,b,c,\alpha,\beta,\gamma}$ satisfy the conditions in Proposition \[pro:semistable\] and suppose $a + b + c + \alpha + \beta + \gamma \equiv 0 \pmod{3}$.

Then there exists a matrix $Z = Z_{a,b,c,\alpha,\beta,\gamma}$ such that

1.  $Z$ is a square integer matrix of size $h_s$,

2.  $R/I$ has the weak Lefschetz property if and only if $\det{Z} \not\equiv 0 \pmod{\operatorname{char}{K}}$, and

3.  the entries of $Z$ are given by $$(Z)_{i,j} = \left\{
                        \begin{array}{ll}
                            1 & n_j \mbox{ is a multiple of } m_i, \\[0.3em]
                            0 & \mbox{otherwise},
                        \end{array}
                    \right.$$ where $\{m_1, \ldots, m_{h_s}\}$ and $\{n_1, \ldots, n_{h_s}\}$ are the monomial 