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In the study of the variety of nilpotent elements in a Lie algebra, Premet conjectured that this variety is irreducible for any finite dimensional restricted Lie algebra. In this paper, with the assumption that the ground field is algebraically closed of characteristic  p  &gt; 3, we confirm this conjecture for the Lie algebras of Cartan type  S ˜  n   and  S n  . Moreover, we show that the variety of nilpotent elements in  S n   is a complete intersection. Motivated by the proof of the irreducibility, we describe explicitly the ring of invariant polynomial functions on  S n  .

The variety of nilpotent elements and invariant polynomial functions on the special algebra S n