When do the symmetric tensors of a commutative algebra form a Frobenius algebra?

For a commutative k-algebra B, consider the subalgebra (B?n)Sn of the nth tensor power of B, formed by the tensors invariant under arbitrary permutations of the indices. Necessary and sufficient conditions are found for (B?n)Sn to be Frobenius. When dimk B $ 2, these say that B is Frobenius and n! is invertible in k, unless B is separable. Some additional cases occur for two-dimensional algebras in positive characteristic, depending on the divisibility of n + 1.