Members’ Seminar

Diagonalization is incredibly important in every field of mathematics. I am a representation theorist, so I will start by motivating the uses of diagonalization in representation theory. Then comes a brief introduction to categorical representation theory, which attempts to lift ideas about (linear transformations acting on vectors in a vector space) to (functors acting on objects in a category). This will be used to motivate and set expectations for a new construction: categorical diagonalization. Suppose you have an operator f whose minimal polynomial factors into linear terms, with distinct roots $k_0$, $k_1$, through $k_r$. Then the projection $p_i$ to the $(k_i)$-th eigenspace can be described as a polynomial in f, using a technique known as Lagrange interpolation. These idempotents equip the vector space with a decomposition into eigenspaces. We think of the process of finding a complete family of orthogonal idempotents $\{p_i\}$ as the diagonalization of $f$. Here is categorical diagonalization: given a functor $F$ with some additional data (akin to the collection of scalars), we construct idempotent functors $P_i$. These can be used to equip the category with a filtration whose subquotients are eigencategories. This is all joint work with Matt Hogancamp. The goal of this talk is not to explain all the details of this construction, but to outline a lot of the key ideas, and give the flavor of the field. We will follow a running example involving modules over the group algebra $A$ of the group of size 2, that is $A = Z[x]/(x^2-1)$. If you know what a short exact sequence of $A$-modules is, you should have all the needed prerequisites for this talk. It also helps to know what a complex of A-modules is, and a homotopy equivalence of such complexes. (If you know about coherent sheaves on projective space, then you'll have a lot of fun ignoring my lecture and thinking about how to diagonalize $O(1)$; this is one topic in a beautiful paper by Gorsky-Negut-Rasmussen.)