holds.

This gives further conditions on the parameters that force the weak Lefschetz property in characteristic zero (see [@BK Theorem 3.3]). This extends [@MMN Lemma 6.7].

\[cor:wlp-ss\] Let $K$ be an algebraically closed field of characteristic zero, and let $I = I_{a,b,c,\alpha,\beta,\gamma}$. If any of the conditions (i)-(iv) in Proposition \[pro:semistable\] fail, then $R/I$ has the weak Lefschetz property.

The above definitions of $s, A, B, C,$ and $M$ are not without purpose. Before going further, we make a few comments about the given parameters.

\[rem:sabcm\] Suppose $s, A, B, C,$ and $M$ are defined as in Proposition \[pro:semistable\]. Then clearly $s$ is an integer if and only if $a + b + c + \alpha + \beta + \gamma \equiv 0 \pmod{3}$; if $s$ is an integer, then so are $A, B, C,$ and $M$. Further, $A+B+C+M = s+2$ and $A+B+C=\alpha + \beta + \gamma$.

 

Associated matrices
-------------------

  Given the minimal free resolution of $R/I$ (see (\[eqn:resolution\])), we can easily compute the $h$-vector of $R/I$ as a weighted sum of binomial coefficients dependent only on the parameters $a,b,c,\alpha, \beta,$ and $\gamma$.

We say $h(A)$ has [*twin peaks*]{} if there exists an integer $s$ such that $h_s = h_{s+1}$. When 