Sadhanandh Vishwanath: An Invitation to Diagrammatic Representation Theory

Abstract: Classical Schur–Weyl Duality, among several equally important results, characterizes the linear maps of GL_n(ℂ) on tensor powers of the standard representation ℂ^n as permutations of tensor indices. This perspective allows us to visualise GL_n(ℂ) linear maps as specific "strand diagrams." For orthogonal and symplectic groups, the corresponding linear maps are represented by specialized diagrams known as Brauer Diagrams, which extends permutation diagrams. A diagrammatic approach enables the study of various representation theoretic properties through the combinatorics of diagrams, which is the central theme of Diagrammatic Representation Theory.

In this talk, I will explain how to encode linear maps using specific diagrams and relate them to elements of certain PROPs. We will explore these concepts within the contexts of the Permutation PROP, Brauer Category and the Partition Category. If time permits, I will also discuss the deformation of such diagrammatic categories.