Symmetric Self-Adjunctions: A Justification of Brauer's Representation of Brauer's Algebras

A classic result of representation theory is Brauer’s construction of a diagrammatical (geometrical) algebra whose matrix representation is a certain given matrix algebra, which is the commutating algebra of the enveloping algebra of the representation of the orthogonal group. The purpose of this paper is to provide a motivation for this result through the categorial notion of symmetric self-adjunction.