Type: Seminar

Distribution: World

Expiry: 10 Oct 2017

CalTitle1: Algebra Seminar: Categorical diagonalization

Auth: kevinc@10.17.192.138 (kcou7211) in SMS-WASM

Ben Elias (University of Oregon) Friday 1 September, 12-1pm, Place: Carslaw 375 Title: Categorical Diagonalization Abstract: 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 Z/2Z. 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.