Algebra Seminar: Elias -- Categorical diagonalization

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.