# Ben Elias (University of Oregon)

## 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.