Defining an Affine Partition Algebra

TL;DR

This paper introduces an affine partition algebra, establishing its foundational properties, its relation to existing algebraic structures, and its connection to the affine partition and Heisenberg categories.

Contribution

It defines a new affine partition algebra, extending Schur-Weyl duality and linking it to the affine partition and Heisenberg categories.

Findings

Defines the affine partition algebra with generators and relations.

Shows the algebra extends Schur-Weyl duality.

Relates the algebra to affine partition and Heisenberg categories.

Abstract

We define an affine partition algebra by generators and relations and prove a variety of basic results regarding this new algebra analogous to those of other affine diagram algebras. In particular we show that it extends the Schur-Weyl duality between the symmetric group and the partition algebra. We also relate it to the affine partition category recently defined by J. Brundan and M. Vargas. Moreover, we show that this affine partition category is a full monoidal subcategory of the Heisenberg category.