Artinian Gorenstein rings with the action of the symmetric group

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We let the symmetric group act on the polynomial ring, and used the Weyl duality to obtain certain results on the invariant coalgebras by Young subgroups of the symmetric group. As results, we obtained the following. (1) We introduced the notion of higher Hessians and using it, we characterized the strong Lefschetz elements in Artinian Gorenstein local rings. (2) We proved that the coalgebras of invariants by the Young subgroups have the strong Lefschetz property. (3) We obtained the irreducible decomposition of the exterior algebra of the differential module over the polynomial ring. (4) We showed some properties of power some symmetric functions in three variables and made a conjecture to the problem to ask in which choice of degrees the power sums form a regular sequence.