Partition algebras

The partition algebra CAk(n) is the centralizer algebra of Sn acting on the k-fold tensor product V⊗k of its n-dimensional permutation representation V. The partition algebra CAk+12(n) is the centralizer algebra of the restriction of V⊗k to Sn−1⊆Sn. We apply the theory of the basic construction (generalized matrix algebras) to the tower of partition algebras CA0(n)⊆CA12(n)CA1(n)⊆CA112(n)⊆⋯. Our main results are: