Partition algebras Pk(n) with 2k > n and the fundamental theorems of invariant theory for the symmetric group Sn

Assume M-n is the n-dimensional permutation module for the symmetric group S-n, and let M-n(circle times k) be its k-fold tensor power. The partition algebra P-k(n) maps surjectively onto the centralizer algebra End(Sn) (M-n(circle times k)) for all k, n is an element of Z(>= 1) and isomorphically when n >= 2k. We describe the image of the surjection Phi(k,n) : P-k(n) -> End(Sn) (M-n(circle times k)) explicitly in terms of the orbit basis of P-k(n) and show that when 2k > n the kernel of Phi(k,n) is generated by a single essential idempotent e(k,n), which is an orbit basis element. We obtain a presentation for End(Sn) (M-n(circle times k)) by imposing one additional relation, e(k,n) = 0, to a presentation of the partition algebra P-k(n) when 2k > n. As a consequence, we obtain the fundamental theorems of invariant theory for the symmetric group S-n. We show under the natural embedding of the partition algebra P-n(n) into P-k(n) for k >= n that the essential idempotent e(n,n) generates the kernel of Phi(k,n). Therefore, the relation e(n,n) = 0 can replace e(k,n) = 0 when k >= n.