Symmetric functions, towers of algebras, and Heisenberg categories

Recent work on the border of Lie theory and low-dimensional topology has lead to a wealth of new algebraic objects with rich combinatorial structure. Unsurprisingly given the setting, symmetric functions appear to play a starring role in this area, seemingly controlling much of the underlying combinatorics. This talk will discuss one example of this phenomenon: the appearance of symmetric functions (and their analogues) in the center of Heisenberg categories. 

 

Each Heisenberg category is attached to a tower of algebras and can be realized by a graphical calculus of string diagrams. The center of the category (the closed diagrams) captures the structure of the centers of all algebras in the tower simultaneously. We will focus on the Heisenberg categories attached to: the group algebras of symmetric groups and algebras of Sergeev superalgebras. We will show how known (inhomogeneous) bases of the symmetric functions appear naturally from the diagrammatics in each case, and how combinatorial questions like the value of structure constants for multiplication of conjugacy class sums can be described diagrammatically. Finally, we will discuss a surprising connection to Markov processes on branching graphs for these towers of algebras.