The weak Lefschetz property of equigenerated monomial ideals

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≥2 and n≥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 Z/dZ, for any n≥3 and any d≥2. We give a complete classification of such ideals in terms of the weak Lefschetz property depending on the action.