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In this note we show that the complete symmetric polynomials are dualgenerators of compressed artinian Gorenstein algebras satisfying the StrongLefschetz Property. This is the first example of an explicit dual form withthese properties.  For complete symmetric forms of any degree in any number of variables, weprovide an upper bound for the Waring rank by establishing an explicit powersum decomposition.  Moreover, we determine the Waring rank, the cactus rank, the resolution andthe Strong Lefschetz Property for any Gorenstein algebra defined by a symmetriccubic form. In particular, we show that the difference between the Waring rankand the cactus rank of a symmetric cubic form can be made arbitrarily large byincreasing the number of variables.  We provide upper bounds for the Waring rank of generic symmetric forms ofdegrees four and five.

Waring and cactus ranks and Strong Lefschetz Property for annihilators of symmetric forms