Partition algebras

The partition algebra C A k ( n ) is the centralizer algebra of S n acting on the k -fold tensor product V ⊗ k of its n -dimensional permutation representation V . The partition algebra C A k + 1 2 ( n ) is the centralizer algebra of the restriction of V ⊗ k to S n − 1 ⊆ S n . We apply the theory of the basic construction (generalized matrix algebras) to the tower of partition algebras C A 0 ( n ) ⊆ C A 1 2 ( n ) C A 1 ( n ) ⊆ C A 1 1 2 ( n ) ⊆ ⋯ . Our main results are:(a) entation on generators and relations for C A k ( n ) ; vation of “Specht modules” from the basic construction; f that C A k ( n ) is semisimple if and only if k ≤ ( n + 1 ) / 2 (except for a few special cases); elements for C A k ( n ) ; and osition on the theory of the basic construction and semisimple algebras.