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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 elementif and only if it is not fixed by any reflections. We also give the necessaryand sufficient condition for strong Lefschetz elements in the invariantsubrings of the coinvariant rings of Weyl groups.

Strong Lefschetz elements of the coinvariant rings of finite Coxeter groups