Algebra Structure On Set Partitions

The partition algebra is an algebra with a basis of set partitions diagrams. Its subalgebra includes diagram algebras such as the uniform block permutations and the group algebra of the symmetric group.

We connect the Hopf algebra of uniform block permutations to the diagram algebra known as the party algebra. This is done by describing a new basis of the partition algebra and looking at the relationship to the basis given for the Hopf algebra of uniform block permutations.

The product and coproduct of the Hopf algebra of uniform block permutations are the generalization of the product of the Malvenuto-Reutenauer Hopf algebra of permutations. We connect the product of the uniform block permutations with the bases of the partition algebra. The centralizer algebra has an internal product and we define an external product on the partition algebra. This algebra contains the algebra of uniform block permutations and the algebra of permutations.