ul 2 01 7 Partition algebras P k ( n ) with 2 k > n and the fundamental theorems of invariant theory for the symmetric group

Assume Mn is the n-dimensional permutation module for the symmetric group Sn, and let M n be its k-fold tensor power. The partition algebra Pk(n) maps surjectively onto the centralizer algebra EndSn(M ⊗k n ) for all k, n ∈ Z≥1 and isomorphically when n ≥ 2k. We describe the image of the surjection Φk,n : Pk(n) → EndSn(M ⊗k n ) explicitly in terms of the orbit basis of Pk(n) and show that when 2k > n the kernel of Φk,n is generated by a single essential idempotent ek,n, which is an orbit basis element. We obtain a presentation for EndSn(M ⊗k n ) by imposing one additional relation, ek,n = 0, to the standard presentation of the partition algebra Pk(n) when 2k > n. As a consequence, we obtain the fundamental theorems of invariant theory for the symmetric group Sn. We show under the natural embedding of the partition algebra Pn(n) into Pk(n) for k ≥ n that the essential idempotent en,n generates the kernel of Φk,n. Therefore, the relation en,n = 0 can replace ek,n = 0 when k ≥ n. 2010 Mathematics Subject Classification 05E10 (primary), 20C30 (secondary).