  We determine a sharp lower bound for the Hilbert function in degree $d$ of a
monomial algebra failing the weak Lefschetz property over a polynomial ring
with $n$ variables and generated in degree $d$, for any $d\geq 2$ and $n\geq
3$. We consider artinian ideals in the polynomial ring with $n$ variables
generated by homogeneous polynomials of degree $d$ invariant under an action of
the cyclic group $\mathbb{Z}/d\mathbb{Z}$, for any $n\geq 3$ and any $d\geq 2$.
We give a complete classification of such ideals in terms of the weak Lefschetz
property depending on the action.
