bases of $[R/I]_s$ and $[R/I]_{s+1}$, respectively, and are given in lexicographic order.

We notice that the map $\times(x+y+z): [R/I]_s \rightarrow [R/I]_{s+1}$ can be represented as a matrix $Z$ with rows and columns indexed by fixed monomial bases of $[R/I]_s$ and $[R/I]_{s+1}$, respectively. This follows immediately from viewing $[R/I]_d$ as a vector space over $K$.

Claim (i) follows from Lemma \[lem:twin-peaks\] wherein it is shown that $h_s = h_{s+1}$. Since $Z$ is square, then the injectivity of $\times(x+y+z): [R/I]_s \rightarrow [R/I]_{s+1}$ is equivalent to $Z$ being invertible, that is, equivalent to $\det{Z}$ being non-zero in $K$. Thus, claim (ii) follows from Corollary \[cor:one-map\] wherein it is shown that the injectivity of the map $\times(x+y+z): [R/I]_s \rightarrow [R/I]_{s+1}$ exactly determines the presence of the weak Lefschetz property for $R/I$. Claim (iii) follows immediately from the construction of the map.

The following generalises the results in [@MMN Theorem 7.2 and Corollary 7.3].

\[pro:wlp-binom\] 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 $N = N_{a,b,c,\alpha,\beta,\gamma}$ such that

1.  $N$ is a square integer matrix of size $C+M$,

2.  