University of Sydney Algebra Seminar

Ben Elias (University of Oregon)

Friday 1 September, 12-1pm, Place: Carslaw 375

Categorical Diagonalization

We know what it means to diagonalize an operator in linear algebra. What might it mean to diagonalize a functor?

Suppose you have an operator f and a collection of distinct scalars ${\kappa }_i$ such that $\prod (f-{\kappa }_i)=0$. Then Lagrange interpolation gives a method to construct idempotent operators $p_i$ which project to the ${\kappa }_i$-eigenspaces of f. We think of this process as diagonalization, and we categorify it: given a functor $F$ with some additional data (akin to the collection of scalars), we construct a complete system of orthogonal idempotent functors $P_i$. We will give some simple but interesting examples involving modules over the group algebra of $\mathbb{Z}/2\mathbb{Z}$. The categorification of Lagrange interpolation is related to the technology of Koszul duality.

Diagonalization is incredibly important in every field of mathematics. I am a representation theorist, so I will briefly indicate some of the important applications of categorical diagonalization to representation theory. Significantly, the "Okounkov-Vershik approach" to the representation theory of the symmetric group can be categorified in this manner. This is all joint work with Matt Hogancamp.