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We give a sharp lower bound for the Hilbert function in degree  d  of artinian quotients    $$\Bbbk [x_1,\ldots ,x_n]/I$$  k [ x 1 , … , x n ] / I failing the Strong Lefschetz property, where  I  is a monomial ideal generated in degree    $$d \ge 2$$  d ≥ 2 . We also provide sharp lower bounds for other classes of ideals, and connect our result to the classification of the Hilbert functions forcing the Strong Lefschetz property by Zanello and Zylinski.

Monomial ideals and the failure of the Strong Lefschetz property