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We introduce the k-strong Lefschetz property (k-SLP) and the k-weak Lefschetz property (k-WLP) for graded Artinian K-algebras, which are generalizations of the Lefschetz properties. The main results obtained in this paper are as follows:1. Let I be a graded ideal of R = K[x 1 , x 2 , x 3 ] whose quotient ring R/I has the SLP. Then the generic initial ideal of I is the unique almost revlex ideal with the same Hilbert function as R/I.2. Let I be a graded ideal of R = K[x 1 , x 2 , . . . , x n ] whose quotient ring R/I has the n-SLP. Suppose that all k-th differences of the Hilbert function of R/I are quasi-symmetric. Then the generic initial ideal of I is the unique almost revlex ideal with the same Hilbert function as R/I.3. We give a sharp upper bound on the graded Betti numbers of Artinian K-algebras with the k-WLP and a fixed Hilbert function.

For the coinvariant rings of finite Coxeter groups of types other than H 4 , we show that a homogeneous element of degree one is a strong Lefschetz element if and only if it is not fixed by any reflections. We also give the necessary and sufficient condition for strong Lefschetz elements in the invariant subrings of the coinvariant rings of Weyl groups.

We prove that the coinvariant ring of the irreducible Coxeter group of type H 4 has the strong Lefschetz property.